Functors on relational structures which admit both left and right adjoints

TL;DR

This paper generalizes the concept of adjoint functors in the homomorphism preorder of relational structures, extending previous work on digraphs, and introduces new constructions relevant to finite duality and promise constraint satisfaction.

Contribution

It extends the theory of adjoint functors to arbitrary relational structures, providing new right adjoints and dual constructions, and connects these to applications in promise constraint satisfaction.

Findings

Generalized adjoint functors to relational structures

Constructed new right adjoints and duals for trees

Linked functor constructions to promise constraint satisfaction

Abstract

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case, $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems.

Equations

$$(f(a_1),\dots ,f(a_{\mathrm{ar}R}))\in R^{\mathbf{B}}.$$

$$f^R((a_1,\dots ,a_{\mathrm{ar}R}))=(f(a_1),\dots ,f(a_{\mathrm{ar}R}))$$

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{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{31.29138pt}{0.0pt}\pgfsys@lineto{53.6069pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.6069pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} {{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}{}{}{}{{}}\pgfsys@moveto{82.51965pt}{0.0pt}\pgfsys@lineto{60.70413pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{60.70413pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\ \operatorname{\sf edge}_{E}(\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}),{\operatorname{\sf vertex}})&\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}))\end{matrix}$$

$$\displaystyle\operatorname{\sf edge}_{E}\bigl{(}\operatorname{\sf pr}_{1}\operatorname{\sf edge}_{E}(\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}),{\operatorname{\sf vertex}}),\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})\bigr{)}$$

$$\operatorname{\sf edge}_{E}\bigl{(}\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},\operatorname{\sf pr}_{1}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})),\operatorname{\sf pr}_{2}\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})\bigr{)}.$$

$$t=\operatorname{\sf edge}_{E}\bigl{(}\operatorname{\sf pr}_{1}(\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})),\operatorname{\sf pr}_{1}(\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}))\bigr{)}$$

$${\text{\&}}^R(e{)}_{{\mathrm{edge}}_R(t_1,\dots ,t_k)}=v_{t_1}^1\wedge \cdots \wedge v_{t_k}^k,$$

$${\text{\&}}^R(f^R(e){)}_t\Rightarrow f(u_i{)}_{{\mathrm{pr}}_i(t)},$$

$$\displaystyle t_{k}=\operatorname{\sf edge}_{E}\left(\operatorname{\sf pr}_{2}\bigl{(}\operatorname{\sf edge}_{E}\bigl{(}\dots\operatorname{\sf pr}_{2}(\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})),\dots,{\operatorname{\sf vertex}}\bigr{)}\bigr{)},{\operatorname{\sf vertex}}\right),$$

$$\dots$$

$$u\in \{\text{true},\text{false}{\}}^{\{s_0,\dots ,s_{k-1}\}}$$

$$h(1)$$

$$h(2)$$

$$⋮$$

$$h(k)$$

$$(\text{true},\dots ,\text{true},\text{false},\dots ,\text{false})$$

$$R^{\Gamma (\mathbf{A})}=\{(r(x_1),\dots ,r(x_k))\mid r:{\mathbf{Q}}_R\to \mathbf{A}\}.$$

$$U\in \prod _{t\in {\mathcal{T}}_V}\mathcal{P}(\mathrm{hom}(\Gamma (\mathbf{T}(t)),\mathbf{B}))$$

$${\text{\&}}^S(E{)}_{{\mathrm{edge}}_S(t_1,\dots ,t_k)}=U_{t_1}^1\dot{\times }\cdots \dot{\times }{U}_{t_k}^k,$$

$$g(u{)}_t=\{f\circ h\mid h:\mathbf{T}(t)\to \mathbf{A},h(r_t)=u\}.$$

$${\text{\&}}^S(g^S(e){)}_t=\{f\circ h\mid h:\mathbf{T}(t)\to \mathbf{A},h^S(r_t)=e\}$$

$$(f(u_1),\dots ,f(u_k))=(fh(x_1),\dots ,fh(x_k))\in R^{\mathbf{B}}$$

$$x_1$$

$$t_E={\mathrm{edge}}_E({\mathrm{pr}}_1({\mathrm{edge}}_E(\mathrm{vertex},\mathrm{vertex})),{\mathrm{pr}}_1({\mathrm{edge}}_E(\mathrm{vertex},\mathrm{vertex})))$$

$$s_0=\mathrm{vertex}$$

$$s_1={\mathrm{pr}}_1(t_1)$$

$${\text{\&}}^E(U,V{)}_{t_1}=U_{s_0}\dot{\times }{V}_{s_0}\text{and}{\text{\&}}^E(U,V{)}_{t_E}=U_{s_1}\dot{\times }{V}_{s_1}$$

$${\text{\&}}^E(U,V{)}_{t_E}=\{(u_1,u_0,v_0,v_1)\mid (u_0,u_1)\in U_{s_1}\text{ and }(v_0,v_1)\in V_{s_1}\}$$

$${\text{\&}}^E(U,V{)}_{t_1}$$

$${\text{\&}}^E(U,V{)}_{t_E}$$

$$A=\{a^{\prime }\mid (a,a^{\prime })\in U_{s_1}\}.$$

$$A\times \{(a,b)\}\times B\subseteq {\text{\&}}^E(U,V{)}_{t_E}\subseteq \mathrm{hom}(\Gamma ({\mathbf{Q}}_E),\mathbf{H}),$$

Full text

Victor Dalmau is supported by the MiCin (PID2019-109137GBC/AEI/10.13039/501100011033) and the Maria de Maeztu Units of Excellence Programme (CEX2021-00195-M). Andrei Krokhin and Jakub Opršal were supported by the UK EPSRC grant EP/R034516/1. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 101034413.

1. Introduction

The study of homomorphisms between (di)graphs and general relational structures plays an important role in combinatorics and theoretical computer science [HN04, FV98, KŽ17]. The importance in theoretical computer science stems, in particular, from the fact that the Constraint Satisfaction Problem (CSP) and its relatives can be cast as the problem of existence of a homomorphism from one relational structure to another. The class of all relational structures of a given signature (e.g., all digraphs) admits the homomorphism preorder, where $\mathbf{A}\leq\mathbf{B}$ for two structure $\mathbf{A}$ and $\mathbf{B}$ if and only if there exists a homomorphism from $\mathbf{A}$ to $\mathbf{B}$. This preorder is interesting in its own right [HN04], and, moreover, important computational problems such as non-uniform CSPs and promise CSPs can be stated in terms of a relative position of an input structure with respect to a fixed structure or a pair of fixed structures. Specifically, the CSP with a template $\mathbf{A}$ (which is a fixed structure) asks whether a given structure $\mathbf{I}$ satisfies $\mathbf{I}\leq\mathbf{A}$ [FV98, KŽ17], and the promise CSP with template $\mathbf{A},\mathbf{B}$ (which is a pair of structures such that $\mathbf{A}\leq\mathbf{B}$) asks to distinguish between the cases $\mathbf{I}\leq\mathbf{A}$ and $\mathbf{I}\not\leq\mathbf{B}$ [AGH17, KO22]; the promise being that input falls into one of the two (disjoint) cases.

A (thin) functor from the class of structures of some signature to the class of structures of a possibly different signature is a mapping that is monotone with respect to the homomorphism preorder. Many well-known examples of constructions in graph theory, such as the arc graph construction, are functors [FT13, FT15]. In this paper we investigate a kind of homomorphism duality of such functors called adjunction — two functors $\Lambda$ and $\Gamma$ form an adjoint pair if $\Lambda(\mathbf{H})$ maps homomorphically to $\mathbf{G}$ if and only if $\mathbf{H}$ maps homomorphically to $\Gamma(\mathbf{G})$ for all relational structures $\mathbf{G},\mathbf{H}$ of the corresponding signatures. In this case, $\Lambda$ is called a left adjoint of $\Gamma$ and $\Gamma$ is a right adjoint of $\Lambda$. If, for some $\Gamma$, there exists such $\Lambda$ we also say that $\Gamma$ has (or admits) a right adjoint.

Adjunction is a core concept of category theory, although we should note that here we work with posetal categories of relational structures, i.e., only existence of a homomorphism matters, rather than with the category of relational structures together with homomorphisms. Adjoints between categories of relational structures in the stricter categorical sense have been completely described by Pultr [Pul70]. Here, we call these functors left and central Pultr functors — a left Pultr functor is a left adjoint to a central Pultr functor. It is clear, that each pair of left and central Pultr functors is adjoint in our sense as well. Nevertheless, there are more functors that admit a right adjoint than left Pultr functors, e.g., the afore-mentioned arc graph construction. In this paper, we ask the question which central Pultr functors (i.e., the functors which admit a left adjoint by Pultr’s characterisation) admit a right adjoint?

A necessary condition for a central Pultr functor to have a right adjoint was given by Foniok and Tardif [FT15]. They also provided explicit constructions of the right adjoints for some of such functors mapping digraphs to digraphs. Loosely speaking, they gave explicit constructions of right adjoints to functors that either do not change the domain (i.e., vertex set) of the digraph, or such that the new domain is the set of edges (arcs) of the input digraph. In the present paper, we extend their results by proving an explicit construction that works with general relational structures. In particular, we provide adjoints to some of the functors between classes of relational structures satisfying the above-mentioned necessary condition, and that either do not change the domain of the structure, or the new domain is one of the relations of the input structures. Hence we prove a direct generalisation of [FT15] to arbitrary relational signatures (see Sections 5 and 6). Furthermore, by composing right adjoints constructed in this way, we obtain more adjunctions than [FT15] even for the digraph case (see Section 7). Finally, we believe that our constructions are more intuitive than those provided by [FT15] — in particular, the above mentioned necessary condition relates adjunction with finite duality, and therefore a construction of a right adjoint is related to a construction of a dual of a tree. Moreover, every finite tree can be built via natural inductive process. Our constructions, and the proofs that they work, reflect this inductive process for certain trees used in the definition of the functor.

2. Preliminaries

We recall some basic definitions and notation. We use the symbol ${\mathscr{P}}(X)$ to denote the power set of a set $X$.

2.1. Structures and homomorphisms

A directed graph can be defined as a pair $\mathbf{G}=(G,E^{\mathbf{G}})$ where $G$ is the set of vertices of $\mathbf{G}$ and $E^{\mathbf{G}}\subseteq G\times G$ a set of edges of $\mathbf{G}$. This is a special case of a relational structure with a single relation of arity $2$ as defined below.

Definition 2.1**.**

A relational signature $\tau$ is a tuple of relational symbols $R,S,\dots$ where each symbol is assigned a positive integer, called an arity and denoted by $\operatorname{ar}R,\operatorname{ar}S,\dots$.

A relational $\tau$-structure is a tuple $\mathbf{A}=(A;R^{\mathbf{A}},S^{\mathbf{A}},\dots)$, where $A$ is a set called the domain (or universe) of $\mathbf{A}$, and $R^{\mathbf{A}}\subseteq A^{\operatorname{ar}R}$, $S^{\mathbf{A}}\subseteq A^{\operatorname{ar}S}$, … are relations on this domain of the corresponding arity.

The relational symbols $R,S,\dots$ in $\tau$ are also referred to as $\tau$-symbols. When no confusion can arise, we say a $\tau$-structure (dropping “relational”), or even simply a structure, when $\tau$ is clear from the context. Two structures of the same signature are said to be similar. We will call elements from the domain of some structure vertices, and tuples in a relation $R$ of some structure $R$-edges, or simply edges if the symbol $R$ is either irrelevant or clear from the context.

In the rest of the paper, we will not use the symbol $V$ as a relational symbol—we restrict its use to refer to the domain of the structure (e.g., in Definition 2.5 and Section 3 below).

Loosely speaking, a homomorphism between two similar structures is a map that preserves relations, e.g., a graph homomorphism would be a map between the two graphs that preserves edges. Formally, a homomorphism is defined as follows.

Definition 2.2**.**

Let $\mathbf{A}$ and $\mathbf{B}$ be two structures of the same signature $\tau$. A homomorphism $f\colon\mathbf{A}\to\mathbf{B}$ is defined to be a mapping $f\colon A\to B$ such that for each relational symbol $R$ in $\tau$ and each $(a_{1},\dots,a_{\operatorname{ar}R})\in R^{\mathbf{A}}$, we have

[TABLE]

We will write $\mathbf{A}\to\mathbf{B}$ if there exists a homomorphism from $\mathbf{A}$ to $\mathbf{B}$. The set of all homomorphisms from $\mathbf{A}$ to $\mathbf{B}$ is denoted by $\hom(\mathbf{A},\mathbf{B})$.

The above definition can be rephrased by saying that the function $f\colon A\to B$ has a coordinate-wise action on each of the relations $R$, i.e., for each relational symbol $R$ in the signature, the expression

[TABLE]

defines a function $f^{R}\colon R^{\mathbf{A}}\to R^{\mathbf{B}}$. We use the symbol $f^{R}$ throughout this paper.

Finally, since we will be extensively working with the homomorphism preorder, this in particular means that we will often work with structures up to homomorphic equivalence—we say that two structures $\mathbf{A}$ and $\mathbf{B}$ are homomorphically equivalent if we have $\mathbf{A}\to\mathbf{B}$ and $\mathbf{B}\to\mathbf{A}$. Such structures would be identified if we followed the standard procedure to turn the homomorphism preorder into a proper partial order. Note that two structures $\mathbf{A}$ and $\mathbf{B}$ are homomorphically equivalent if and only if for every structure $\mathbf{C}$, we have $\mathbf{C}\to\mathbf{A}$ if and only if $\mathbf{C}\to\mathbf{B}$, i.e., they allow homomorphisms from the same structures. The same is also true for allowing homomorphisms to the same structures, i.e., $\mathbf{A}$ and $\mathbf{B}$ are homomorphically equivalent if and only if, for all $\mathbf{C}$, we have $\mathbf{A}\to\mathbf{C}$ if and only if $\mathbf{B}\to\mathbf{C}$. A structure is called a core, if it is not homomorphically equivalent to any of its proper substructures.

Certain structures called trees play a special role in this paper. We use a definition of a tree equivalent to the one given in [NT00, Section 3]. Loosely speaking, a relational structure is a tree if it is connected, contains no cycles, and none of its relations has tuples with repeated entries. This is more precisely defined by using the incidence graph of a structure.

Definition 2.3**.**

The incidence graph of a structure $\mathbf{A}=(A;R^{\mathbf{A}},\dots)$ is a bipartite multigraph whose vertex set is the disjoint union of $A$ and $R^{\mathbf{A}}$ for each relational symbol $R$. There is an edge connecting every tuple $(a_{1},\dots,a_{k})\in R^{\mathbf{A}}$ with every one of its coordinates $a_{i}\in A$. In particular, if some element appears multiple times in this tuple, then the edge connecting it to the tuple appears with the same multiplicity. In this case the incidence graph contains a cycle of length 2 (i.e., two parallel edges).

A $\tau$-structure is a ($\tau$-)tree if its incidence graph is a tree, i.e., an acyclic connected digraph.

Remark 2.4**.**

An undirected graph can be viewed as a relational structure with a single relation $E$ whose universe is the set of vertices $V$. The relation $E\subseteq V\times V$ contains for every edge two tuples $(u,v)$ and $(v,u)$. This means that no undirected graph with at least one edge is a tree according to the above definition since if $(u,v)$ is an edge, then $u$, $(u,v)$, $v$, $(v,u)$ is a 4-cycle of the incidence graph. Intuitively, relational structures with binary relations are directed graphs; an undirected graph is encoded as directed by including both orientations of each edge which results in a directed cycle of length 2. Under the above definition a directed graph is a tree if it is an oriented tree.

We fix notation for certain small structures containing either a single vertex or a single $R$-edge for some relational symbol $R$ that we will use later.

Definition 2.5**.**

Fix a relational signature $\sigma$. We define the following structures:

• •

$\mathbf{V}_{1}$* is the structure with a single vertex, i.e., $V_{1}=\{1\}$, and empty relations, i.e., $R^{\mathbf{V}_{1}}=\emptyset$ for all $\sigma$-symbols $R$.*

• •

Let $S$ be a relational symbol, $\mathbf{S}_{1}$ is a structure with $\operatorname{ar}S$ vertices related by $S$ and all other relations empty. More precisely, $S_{1}=\{1,\dots,k\}$ where $k=\operatorname{ar}_{S}$, $S^{\mathbf{S}_{1}}=\{(1,\dots,k)\}$, and $R^{\mathbf{S}_{1}}=\emptyset$ for all $\sigma$-symbols $R$ except $S$.

2.2. Pultr functors: gadget replacement and pp-constructions

Traditionally, in CSP literature, pp-constructions would be described in the language of logic using so called primitive positive formulae (logical formulae that use only $\exists$, $\wedge$, and $=$). We refer to [BKW17, Definition 19] for details. In this paper, we define “pp-constructions” using a language similar to [FT15, Definitions 2.1–2.3].

Definition 2.6**.**

Let $\sigma$ and $\tau$ be two relational signatures. A $(\sigma,\tau)$-Pultr template is a tuple of $\sigma$-structures consisting of $\mathbf{P}$, and $\mathbf{Q}_{R}$, one for each $\tau$-symbol $R$, together with homomorphisms $\epsilon_{i,R}\colon\mathbf{P}\to\mathbf{Q}_{R}$ for each $\tau$-symbol $R$ and all $i\in\{1,\dots,\operatorname{ar}R\}$.

Definition 2.7**.**

Given a $(\sigma,\tau)$-Pultr template as above, we define two functors $\Lambda$ and $\Gamma$ called (left and central) Pultr functors.

• •

Given a $\tau$-structure $\mathbf{A}$, we define a $\sigma$-structure $\Lambda(\mathbf{A})$ in the following way: For each $a\in A$, introduce to $\Lambda(\mathbf{A})$ a copy of $\mathbf{P}$ denoted by $\mathbf{P}_{a}$, and for each $\tau$-symbol $R$ and each $(a_{1},\dots,a_{k})\in R^{\mathbf{A}}$, introduce to $\Lambda(\mathbf{A})$ a copy of $\mathbf{Q}_{R}$ with the image of $\mathbf{P}$ under $\epsilon_{i,R}$ identified with $\mathbf{P}_{a_{i}}$ for all $i\in\{1,\dots,k\}$.

• •

Given a $\sigma$-structure $\mathbf{B}$, we define a $\tau$-structure $\Gamma(\mathbf{B})$ whose universe consists of all homomorphisms $h\colon\mathbf{P}\to\mathbf{B}$. The relation $R^{\Gamma(\mathbf{B})}$ where $R$ is a $\tau$-symbol is then defined to contain all tuples $(h_{1},\dots,h_{k})$ of such homomorphisms for which there is a homomorphism $g\colon\mathbf{Q}_{R}\to\mathbf{B}$ such that $h_{i}=g\circ\epsilon_{i,R}$ for all $i\in\{1,\dots,k\}$.

Once the definitions are settled, it is not too hard to show that for any Pultr template, the corresponding Pultr functors $\Lambda$ and $\Gamma$ are left and right adjoints. This statement is attributed to [Pul70] though it was certainly rediscovered on numerous occasions.

The fact that both $\Lambda$ and $\Gamma$ are (thin) functors follows from the adjunction. It can be also easily proved directly, for example, a homomorphism $f^{\Gamma}\colon\Gamma(\mathbf{A})\to\Gamma(\mathbf{B})$ can be obtained from a homomorphism $f\colon\mathbf{A}\to\mathbf{B}$ by setting $f^{\Gamma}(h)=f\circ h$ for each $h\colon\mathbf{P}\to\mathbf{A}$.

Example 2.8**.**

Let us consider the Pultr template consisting of structures $\mathbf{P}=(\{0,1\};\{(0,1)\})$ and $\mathbf{Q}_{E}=(\{0,1,2\};\{(0,1),(1,2)\})$ with $\epsilon_{1}(0)=0$ and $\epsilon_{1}(1)=1$, and $\epsilon_{2}(0)=1$ and $\epsilon_{2}(1)=2$.

The corresponding central Pultr functor $\Gamma$ is the arc-graph construction that is usually denoted by $\delta$; given a (directed) graph $\mathbf{G}=(G,E^{\mathbf{G}})$, the digraph $\delta(\mathbf{G})$ is defined as the graph with the vertex set $E^{\mathbf{G}}$ and edges $((u,v),(v,w))\in E^{\mathbf{G}}\times E^{\mathbf{G}}$.

The left Pultr functor $\Lambda$ corresponding to this template provides a left adjoint to the arc-graph construction, and can be explicitly described as follows: Given a digraph $\mathbf{G}$, the digraph $\Lambda(\mathbf{G})$ is constructed by replacing each vertex $v\in G$ with a pair of vertices $v_{0},v_{1}$ connected by an edge, i.e., with $(v_{0},v_{1})\in E^{\Lambda(\mathbf{G})}$. Furthermore, for each edge $(u,v)\in E^{\mathbf{G}}$, we identify vertices $u_{1}$ and $v_{0}$. For example, the image of a path of length $n$, i.e., the digraph $\mathbf{P}_{n}=(\{0,\dots,n\};\{(i,i+1)\mid i\in\{0,n-1\})$ is the path of length $n+1$ consisting of vertices $0_{0},0_{1}=1_{0},1_{1}=2_{0},\dots,n_{1}$.

Observe that, in this case, we have that $\Lambda(\Gamma(\mathbf{P}_{n}))$ is isomorphic to $\mathbf{P}_{n}$. This does not need to happen for any digraph in place of $\mathbf{P}_{n}$, nevertheless, the existence of a homomorphism $\Lambda(\Gamma(\mathbf{G}))\to\mathbf{G}$, for all digraphs $\mathbf{G}$, follows from the adjunction in a straightforward way.

We remark that the connection between the adjunction of left and central Pultr functors and the algebraic reductions between CSPs has been described in [KOWŽ23, Section 4.1]. Let us briefly mention that the central Pultr functors are pp-constructions (more precisely, a structure $\mathbf{A}$ is pp-constructible from $\mathbf{B}$ if it is homomorphically equivalent to the structure $\Gamma(\mathbf{B})$ for some Pultr functor $\Gamma$), and the left Pultr functors are called gadget replacements in this context.

The main contribution of the present paper is an investigation of the cases of Pultr templates for which $\Gamma$ is also a left adjoint, i.e., it admits a right adjoint $\Omega$. The following necessary condition for this was proved in [FT15] for the case of digraphs, but the proof goes through verbatim for general structures; we include it for completeness.

Theorem 2.9** ([FT15, Theorem 2.5]).**

Let $\Lambda$ and $\Gamma$ be a pair of left and central Pultr functors defined by a $(\sigma,\tau)$-Pultr template. If $\Gamma$ has a right adjoint, then, for each $\tau$-tree $\mathbf{T}$, $\Lambda(\mathbf{T})$ is homomorphically equivalent to a $\sigma$-tree.

Proof 2.10**.**

A pair $(\mathbf{T},\mathbf{D})$ of similar structures is called a duality pair if, for all structures $\mathbf{A}$ similar to $\mathbf{T}$, either $\mathbf{T}\to\mathbf{A}$ or $\mathbf{A}\to\mathbf{D}$. In this case, $\mathbf{D}$ is called a dual to $\mathbf{T}$. It was shown in [NT00] that a structure has a dual if and only if this structure is homomorphically equivalent to a tree.

Thus, it suffices to prove that if $\Gamma$ has both a left adjoint $\Lambda$ and a right adjoint $\Omega$, and $(\mathbf{T},\mathbf{D})$ is a duality pair then $(\Lambda(\mathbf{T}),\Omega(\mathbf{D}))$ is a duality pair as well. To show that, observe that $\Lambda(\mathbf{T})\not\to\mathbf{A}$ is equivalent to $\mathbf{T}\not\to\Gamma(\mathbf{A})$ since $\Lambda$ is a left adjoint to $\Gamma$. The latter condition is equivalent to $\Gamma(\mathbf{A})\to\mathbf{D}$, since $\mathbf{D}$ is a dual of $\mathbf{T}$, which in turn is equivalent to $\mathbf{A}\to\Omega(\mathbf{D})$, since $\Omega$ is a right adjoint to $\Gamma$.

A simple corollary of the above is that if a central Pultr functor $\Gamma$ has a right adjoint, then all the structures in its Pultr template are homomorphically equivalent to trees. This is because, as it is shown in [FT15] (and not hard to see), if the template consists of structures $\mathbf{P}$ and $\mathbf{Q}_{R}$ for $\tau$-symbols $R$, then $\mathbf{P}=\Lambda(\mathbf{V}_{1})$ and $\mathbf{Q}_{R}=\Lambda(\mathbf{R}_{1})$ for each $\tau$-symbol $R$, where (tree) $\tau$-structures $\mathbf{V}_{1}$ and $\mathbf{R}_{1}$ are as in Definition 2.5.

It is open whether the above condition is also sufficient to have a right adjoint. In this paper, as well as Foniok-Tardif do in [FT15], we focus on the cases when $\mathbf{P}$ and $\mathbf{Q}_{R}$’s are actually trees, and we give a concrete construction of the adjoint in two cases: $\mathbf{P}=\mathbf{V}_{1}$ (this is shown in Section 5), and $\mathbf{P}=\mathbf{S}_{1}$ is the tree with a single $S$-edge for some $\sigma$-symbol $S$ (this is shown in Section 6). Finally, in the last section we combine these two constructions to prove that if $\mathbf{P}$ and $\mathbf{Q}_{R}$’s are trees, and moreover the images of $\mathbf{P}$ in $\mathbf{Q}_{R}$ under the maps $\epsilon_{i,R}$ are either disjoint or intersect in one vertex, then the corresponding central Pultr functor has a right adjoint (Theorem 7.5).

Finally, let us note a well-known fact from category theory that if $\Omega_{1}$ and $\Omega_{2}$ are both right adjoints to $\Gamma$, then $\Omega_{1}(\mathbf{A})$ and $\Omega_{2}(\mathbf{A})$ are homomorphically equivalent for all structures $\mathbf{A}$. The proof is immediate from the definitions: both structures allow a homomorphism from $\mathbf{B}$ if and only if $\Gamma(\mathbf{B})\to\mathbf{A}$.

3. An inductive construction of trees

For our constructions, it will be convenient to describe $\tau$-trees by certain formal terms. Our terms will correspond to trees that are rooted, either in a vertex or in an edge (i.e., a tuple in one of the relations). Each term will correspond to a unique inductive construction of a tree, but the same tree can be obtained by several inductive constructions.

We recall that we assume that none of the relational signatures uses $V$ as a relational symbol. Each term will be either a $V$-term ($V$ stands for vertices) or an $R$-term where $R$ is a relational symbol in $\tau$. The rules for the inductive construction of terms and their semantics are as follows:

• •

${\operatorname{\sf vertex}}$ is a $V$-term.

• •

If $t_{1},\ldots,t_{k}$ are $V$-terms and $R$ is a $k$-ary symbol in $\tau$ then $\operatorname{\sf edge}_{R}(t_{1},\ldots,t_{k})$ is an $R$-term.

• •

If $t$ is an $R$-term and $i\in\{1,\dots,\operatorname{ar}R\}$, then $\operatorname{\sf pr}_{i}(t)$ is a $V$-term.

The rooted tree $({\mathbf{T}(t)},r_{t})$, where $r_{t}$ is the root, corresponding to a term $t$ is defined in the following way:

• •

${\mathbf{T}({\operatorname{\sf vertex}})}$ is the one-vertex tree, rooted at its only vertex, i.e., ${\mathbf{T}({\operatorname{\sf vertex}})}=\mathbf{V}_{1}$ and $r_{t}=1$.

• •

If $t=\operatorname{\sf edge}_{R}(t_{1},\ldots,t_{k})$ is an $R$-term then ${\mathbf{T}(t)}$ is the tree obtained by taking the disjoint union of the trees ${\mathbf{T}(t_{i})}$ (with roots $r_{t_{i}}$) and adding the tuple $(r_{t_{1}},\dots,r_{t_{k}})$ to the relation $R$. The root $r_{t}$ is defined to be this new tuple.

• •

If $t=\operatorname{\sf pr}_{i}(s)$ is a $V$-term and $r_{s}=(v_{1},\dots,v_{k})$, then we set ${\mathbf{T}(t)}={\mathbf{T}(s)}$ and $r_{t}=v_{i}$.

We say that $t$ represents a tree $\mathbf{T}$ if ${\mathbf{T}(t)}$ is isomorphic to $\mathbf{T}$, and that $t$ represents a tree $\mathbf{T}$ rooted in $r$ if ${\mathbf{T}(t)}$ is isomorphic to $\mathbf{T}$ via an isomorphism mapping $r_{t}$ to $r$. The same tree can in general be represented by several terms (see the following example), but each term represents one tree.

Example 3.1**.**

The following are examples of terms and the trees they represent in the relational signature of digraphs. The corresponding roots are highlighted. We drop the brackets around arguments of $\operatorname{\sf pr}_{i}$ to ease readability.

[TABLE]

Note that the difference between the second tree and the third tree is the root; the latter tree is obtained from the other by applying $\operatorname{\sf pr}_{2}$, and thus only changing the root. Further, we present a few examples of more complicated trees.

[TABLE]

We remark that a tree can be represented by multiple terms even when we fix the root. For example, the last tree above can be also represented by the term

[TABLE]

Lemma 3.2**.**

Fix a relational signature $\tau$. Any finite $\tau$-tree can be represented by a term. More precisely,

• •

for every finite tree $\mathbf{T}$ and $r\in T$ there is a $V$-term $t$ such that $\mathbf{T}(t)$ is isomorphic to $\mathbf{T}$ via an isomorphism that maps $r_{t}$ to $r$, and

• •

for every finite tree $\mathbf{T}$ and $r\in R^{\mathbf{T}}$ for some $\tau$-symbol $R$, there is an $R$-term $t$ such that $\mathbf{T}(t)$ is isomorphic to $\mathbf{T}$ via an isomorphism that maps $r_{t}$ to $r$.

The proof is a simple argument by induction on the number of edges of the tree. We include it in detail to provide more intuition about terms.

Proof 3.3**.**

We prove this statement by an induction on the number of edges of $\mathbf{T}$. Each induction step is moreover split in two: we first prove that trees with $n$ edges rooted in an edge can be represented, and then, assuming the above, we show that trees with $n$ edges rooted in a vertex can be represented.

1. (1)

We start with $n=0$. There is a single tree $\mathbf{T}$ with no edges, namely the tree with one vertex. It cannot be rooted in an edge, but it can be rooted in its only vertex. It is represented by the term ${\operatorname{\sf vertex}}$. 2. (2)

Assume $n>0$, and we can represent all trees rooted in a vertex with less than $n$ edges. Assume that $\mathbf{T}$ has $n$ edges and is rooted in an edge $(r_{1},\dots,r_{k})\in R^{\mathbf{T}}$ for some $k$-ary symbol $R$. Removing this edge from $\mathbf{T}$ splits $\mathbf{T}$ into $k$ connected components $\mathbf{T}_{1},\dots,\mathbf{T}_{k}$ where $\mathbf{T}_{i}$ contains $r_{i}$ for each $i$. Each of these components is a tree. Assuming that $\mathbf{T}_{i}$ rooted in $r_{i}$ is represented by the term $t_{i}$ for each $i$, $\mathbf{T}$ rooted in $(r_{1},\dots,r_{k})$ is represented by $\operatorname{\sf edge}_{R}(t_{1},\dots,t_{k})$. 3. (3)

Assume that $n>0$, and we can represent all trees rooted in an edge with at most $n$ edges. Let $\mathbf{T}$ be a tree with $n$ edges and $r\in T$. Since $\mathbf{T}$ is connected, $r$ is involved in some edge, say $(v_{1},\dots,v_{k})\in R^{\mathbf{T}}$ where $k=\operatorname{ar}R$ and $r=v_{i}$ for some $i$. Now, $\mathbf{T}$ rooted in $(v_{1},\dots,v_{k})$ is represented by an $R$-term $t$ by the inductive assumption, so $\mathbf{T}$ rooted in $r$ is represented by $\operatorname{\sf pr}_{i}(t)$.

Finally, our constructions use the notion of a subterm of a term. Intuitively a subterm of a term $t$ is any proper term that appears as a part of $t$. The set of all subterms of $t$ encodes all intermediate byproducts of the inductive construction of ${\mathbf{T}(t)}$. Formally, we define subterms as follows.

• •

The only subterm of ${\operatorname{\sf vertex}}$ is itself.

• •

If $t=\operatorname{\sf edge}_{R}(t_{1},\ldots,t_{k})$ is an $R$-term where $t_{1},\ldots,t_{k}$ are $V$-terms and $R$ is a $k$-ary symbol in $\tau$ then the set of its subterms consists of the term itself and all subterms of $t_{1},\ldots,t_{k}$.

• •

If $t=\operatorname{\sf pr}_{i}(s)$ is a $V$-term where $s$ is an $R$-term, then the set of its subterms consists of the term itself and all subterms of $s$.

If $s$ is a subterm of $t$, then we write $s\leq t$, moreover if $s$ is a proper subterm of $t$, i.e., it is a subterm and $s\neq t$, we write $s<t$. A $V$-subterm of a term $t$ is a subterm which is a $V$-term, and similarly an $R$-subterm for a relational symbol $R$ is a subterm which is an $R$-term.

For example, the term

[TABLE]

has four distinct subterms: two $E$-terms, which are $t$ and $\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})$, and two $V$-terms, which are $\operatorname{\sf pr}_{1}(\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}))$ and ${\operatorname{\sf vertex}}$.

We note that statements about terms can be proven by an inductive principle: showing the statement is true for the term ${\operatorname{\sf vertex}}$, and then showing that if it is true for all proper subterms of a term $t$, it is also true for $t$.

4. Prelude: Duals to trees

Before we get to the main construction of adjoint, let us briefly discuss a simpler construction of a dual of a tree. There are a few similarities between the construction of duals and right adjoints to central Pultr functors: as we mentioned before (see the proof of Theorem 2.9), if $\Gamma$ is a central Pultr functor that has a left adjoint $\Lambda$ and a right adjoint $\Omega$, and $(\mathbf{T},\mathbf{D})$ is a duality pair, then $(\Lambda(\mathbf{T}),\Omega(\mathbf{D}))$ is also a duality pair. Moreover, our construction of the dual uses the inductive construction of trees from the previous section in a similar way as the constructions of right adjoints in Sections 5 and 6, but the construction of a dual is conceptually easier, so we present it to create some intuition that will be useful below.

Again, we fix a relational signature. Recall that a pair $(\mathbf{T},\mathbf{D})$ of similar structures is called a duality pair if, for all structures $\mathbf{A}$ similar to $\mathbf{T}$, either $\mathbf{T}\to\mathbf{A}$ or $\mathbf{A}\to\mathbf{D}$. In this case, $\mathbf{D}$ is called a dual to $\mathbf{T}$, and it was shown in [NT00] that a structure has a dual if and only if it is homomorphically equivalent to a tree. See [LLT07, NT05] for other characterisations of finite duality. Our construction is loosely inspired by the construction in [NT05].

Another way to look at duals is that for any structure $\mathbf{A}$, a homomorphism $\mathbf{A}\to\mathbf{D}$ should correspond to a ‘proof’ that $\mathbf{T}\not\to\mathbf{A}$. How can one prove that a tree does not map to a structure $\mathbf{A}$ if that is the case? This can be done by an inductive argument. Let us outline this argument in the case $\mathbf{T}$ and $\mathbf{A}$ are digraphs. More precisely, we describe a procedure that shows that for a fixed root $r\in T$ and some $a\in A$, there is no homomorphism from $\mathbf{T}$ to $\mathbf{A}$ that maps $r$ to $a$. Pick a neighbour $s$ of $r$ in $\mathbf{T}$, and assume that $(r,s)\in E^{\mathbf{T}}$, the other orientation is dealt with symmetrically. We can show that there is no homomorphism from $\mathbf{T}$ to $\mathbf{A}$ mapping $r$ to $a$ by showing that for none of the neighbours $b$ of $a$ in $\mathbf{A}$, there is a homomorphism $\mathbf{T}\to\mathbf{A}$ that maps the edge $(r,s)$ to $(a,b)$. In turn, removing the edge $(r,s)$ from $\mathbf{T}$ splits the tree into two subtrees $\mathbf{T}_{1}$, containing $r$, and $\mathbf{T}_{2}$, containing $s$. A homomorphism $\mathbf{T}\to\mathbf{A}$ that maps $(r,s)$ to $(a,b)$ is equivalent to a pair of homomorphisms $\mathbf{T}_{1}\to\mathbf{A}$, that maps $r$ to $a$, and $\mathbf{T}_{2}\to\mathbf{A}$, that maps $s$ to $b$. We have thus reduced the claim to proving that there is no homomorphism from two smaller trees, and can therefore recursively repeat our strategy for these two smaller trees. We further design a structure $\mathbf{D}$ in which we can encode proofs of the above form. In particular, the image of an element $a\in\mathbf{A}$ under a homomorphism $\mathbf{A}\to\mathbf{D}$ will contain answers to questions of the form ‘Is there a homomorphism $\mathbf{T}^{\prime}\to\mathbf{A}$ that maps $r$ to $a$?’ for all trees $\mathbf{T}^{\prime}$ and all roots $r\in T^{\prime}$ that would appear in the above inductive argument.

Definition 4.1**.**

Let $t_{Q}$ be an $S$-term for some relational symbol $S$. Let $\mathscr{T}_{V}$ be the set of all $V$-subterms of $t_{Q}$, and $\mathscr{T}_{R}$ be the set of all $R$-subterms of $t_{Q}$ for each relational symbol $R$.

We define a structure $\mathbf{D}(t_{Q})$. The domain of this structure, denoted by $D(T_{Q})$ is the set of all tuples $v\in\{\textsf{true},\textsf{false}\}^{\mathscr{T}_{V}}$, indexed by $V$-subterms of $t_{Q}$, such that $v_{{\operatorname{\sf vertex}}}=\textsf{true}$. A tuple $e=(v^{1},\dots,v^{k})$ of vertices is related in a relation $R$ (of arity $k$) in $\mathbf{D}(t_{Q})$ if the tuple $\operatorname{\text{\sf\&}}^{\!R}(e)\in\{\textsf{true},\textsf{false}\}^{\mathscr{T}_{R}}$ defined by

[TABLE]

(note that every $R$-term is of the form $\operatorname{\sf edge}_{R}(t_{1},\dots,t_{k})$) satisfies:

1. (D1)

For all $i\in\{1,\dots,k\}$ and $t\in\mathscr{T}_{R}$, if $\operatorname{\sf pr}_{i}(t)\in\mathscr{T}_{V}$, then $\operatorname{\text{\sf\&}}^{\!R}(e)_{t}\Rightarrow v^{i}_{\operatorname{\sf pr}_{i}(t)}$. 2. (D2)

If $R=S$, i.e., if $t_{Q}$ is an $R$-term, then $\operatorname{\text{\sf\&}}^{\!R}(e)_{t_{Q}}=\textsf{false}$.

Item (D1) is essentially quantified by the $V$-subterms of $t_{Q}$ since all of such subterms $t^{\prime}$, with the exception $t^{\prime}={\operatorname{\sf vertex}}$, are of the form $\operatorname{\sf pr}_{i}(t)$ for some $t$, and $t$ and $i$ are uniquely defined by $t^{\prime}$. Below, we use the symbol $\operatorname{\text{\sf\&}}^{\!R}(e)$ for the tuple defined by (1).

Theorem 4.2**.**

For any relational structure $\mathbf{Q}$ that is homomorphically equivalent to $\mathbf{T}(t_{Q})$ for some $S$-term $t_{Q}$ where $S$ is a relational symbol, $\mathbf{D}(t_{Q})$ is a dual of $\mathbf{Q}$.

Proof 4.3**.**

Without loss of generality, we may assume that $\mathbf{Q}=\mathbf{T}(t_{Q})$. We need to show that for all structures $\mathbf{A}$, $\mathbf{A}\to\mathbf{D}(t_{Q})$ if and only if $\mathbf{Q}\not\to\mathbf{A}$.

First, assume that $\mathbf{Q}\not\to\mathbf{A}$. We define $f\colon A\to D(t_{Q})$ by putting, for each $u\in A$ and each $t\in\mathscr{T}_{V}$, $f(u)_{t}=\textsf{true}$ if there is a homomorphism $h\colon\mathbf{T}(t)\to\mathbf{A}$ that maps the root to $u$, and $f(u)_{t}=\textsf{false}$ otherwise. Clearly, $f(u)_{{\operatorname{\sf vertex}}}=\textsf{true}$. Now, assume that $e=(u_{1},\dots,u_{k})\in R^{\mathbf{A}}$. Observe that $\operatorname{\text{\sf\&}}^{\!R}(f^{R}(e))_{t}$ is true if and only if there is a homomorphism $h\colon\mathbf{T}(t)\to\mathbf{A}$ mapping the root edge to $e$. In other words, there is a homomorphism $h\colon\mathbf{T}(t)\to\mathbf{A}$ mapping the root edge to $e$ if and only if there are homomorphisms $h_{i}\colon\mathbf{T}(t_{i})\to\mathbf{A}$ with $h_{i}(r_{t_{i}})=u_{i}$ for all $i$. Assuming that such a homomorphism $h$ exists, the homomorphisms $h_{i}$ are defined as restrictions of $h$ to the corresponding subtrees. Assuming homomorphisms $h_{i}$, taking the union of $h_{i}$ defines a mapping $h$ on all vertices of $\mathbf{T}(t)$. This mapping clearly preserves all edges different from the root since $h_{i}$ are homomorphisms, and it also preserves the root edge, since it is mapped to $e\in R^{\mathbf{A}}$. This means that $h$ is indeed a homomorphism. We need to show that $\operatorname{\text{\sf\&}}^{\!R}(f^{R}(e))$ satisfies (D1) and (D2). For (D1), we want

[TABLE]

i.e., if there is a homomorphism $\mathbf{T}(t)\to\mathbf{A}$ mapping the root to $e$ then there is a homomorphism $\mathbf{T}(\operatorname{\sf pr}_{i}(t))\to\mathbf{A}$ mapping the root to $u_{i}$. This is trivial since $\mathbf{T}(t)=\mathbf{T}(\operatorname{\sf pr}_{i}(t))$ and a homomorphism $h\colon\mathbf{T}(t)\to\mathbf{A}$ that maps the root $r_{t}$ to $(u_{1},\dots,u_{k})$ necessarily maps $r_{\operatorname{\sf pr}_{i}(t)}$ to $u_{i}$. Finally, (D2) is clear from the definition since we assumed that $\mathbf{Q}\not\to\mathbf{A}$.

For the other implication, we first prove the following by induction on the term $t\leq t_{Q}$:

Claim 1**.**

Let $t\leq t_{Q}$ and let $h\colon\mathbf{T}(t)\to\mathbf{D}(t_{Q})$ be a homomorphism. Then $h(r_{t})_{t}=\textsf{true}$ if $t$ is a $V$-term, and $\operatorname{\text{\sf\&}}^{\!R}(h^{R}(r_{t}))_{t}=\textsf{true}$ if $t$ is an $R$-term.

1. (1)

The case $t={\operatorname{\sf vertex}}$ is trivial. 2. (2)

Let $t=\operatorname{\sf edge}_{R}(t_{1},\dots,t_{\operatorname{ar}R})$. We assume that $h\colon\mathbf{T}(t)\to\mathbf{A}$ is a homomorphism and $h(r_{t})=(v^{1},\dots,v^{\operatorname{ar}R})$. This in particular means that for each $i$, $h$ maps the root of $\mathbf{T}(t_{i})$ to $v^{i}$. Hence, we can apply the inductive assumption on the restrictions of $h$ to $\mathbf{T}(t_{i})$’s to get that $v^{i}_{t_{i}}=\textsf{true}$ for all $i$, and consequently the claim by the definition of $\operatorname{\text{\sf\&}}^{\!R}(h^{R}(r_{t}))_{t}$. 3. (3)

Let $t=\operatorname{\sf pr}_{i,R}(t^{\prime})$, and $h\colon\mathbf{T}(t^{\prime})\to\mathbf{A}$. Note that $\mathbf{T}(t)=\mathbf{T}(t^{\prime})$, so $h$ is also a homomorphism from $\mathbf{T}(t)$. Let $h^{R}(r_{t^{\prime}})=(v^{1},\dots,v^{k})$, and observe that $h(r_{t})=v^{i}$. By the inductive assumption, this implies that $\operatorname{\text{\sf\&}}^{\!R}(f^{R}(e))_{t}=\textsf{true}$. Consequently, we get that $h(r_{t})_{t}=v^{i}_{t}=\textsf{true}$ from (D1).

This concludes the proof of the claim. Assume for a contradiction that $f\colon\mathbf{A}\to\mathbf{D}(t_{Q})$ and $g\colon\mathbf{Q}\to\mathbf{A}$ are homomorphisms. Since $\mathbf{T}(t_{Q})=\mathbf{Q}$, the above claim applied to $t=t_{Q}$ and the homomorphism $h=f\circ g\colon\mathbf{Q}\to\mathbf{D}(t_{Q})$, would imply that $\operatorname{\text{\sf\&}}^{\!R}(h^{R}(r_{t_{Q}}))_{t_{Q}}=\textsf{true}$ which would contradict (D2).

4.1. Example: Dual to a directed path

It is well known (see, e.g., [HN04, Proposition 1.20]) that a directed graph maps homomorphically to the graph $\mathbf{L}_{k}=(\{1,\dots,k\};<)$, where the edge relation is given by the strict order on the domain, if and only if it does not allow a homomorphism from a directed path with $k$ edges (and $k+1$ vertices) — we denote this path by $\mathbf{P}_{k}$. This means that $\mathbf{L}_{k}$ is the dual of $\mathbf{P}_{k}$. Let us compare this observation with our construction of the dual.

Fix $k>0$. We pick the term

[TABLE]

where $\operatorname{\sf edge}_{E}$ appears $k$ times, to represent $\mathbf{P}_{k}$ rooted in its last edge, i.e., the following graph.

[TABLE]

We also let $s_{0}={\operatorname{\sf vertex}}$, $t_{i}=\operatorname{\sf edge}_{E}(s_{i-1},{\operatorname{\sf vertex}})$ and $s_{i}=\operatorname{\sf pr}_{2}(t_{i})$ for $i\in\{1,\dots,k-1\}$. Note that both $t_{i}$ and $s_{i}$ represent a path of length $i$ — the difference is the root which is either the last edge or the last vertex. Finally note that $s_{0}$, …, $s_{k-1}$ and $t_{1}$, …, $t_{k}$ are the only subterms of $t_{k}$.

By definition, vertices of $\mathbf{D}(t_{k})$ are tuples

[TABLE]

such that $u_{s_{0}}=\textsf{true}$. This allows us to write them simply as ordered $k$-tuples whose $i$-th entry is the value corresponding to $s_{i-1}$. Now, $u$ is connected by an edge to $v$ if

1. (D1)

$\operatorname{\text{\sf\&}}^{\!E}(u,v)_{t_{i}}\Rightarrow v_{s_{i}}$ for all $i$ since $s_{i}=\operatorname{\sf pr}_{2}(t_{i})$ (the implication $e_{t_{i}}\Rightarrow v_{\operatorname{\sf pr}_{1}(t_{i})}$ is not needed since $\operatorname{\sf pr}_{1}(t_{i})$ is not a subterm of $t_{k}$), and 2. (D2)

$\operatorname{\text{\sf\&}}^{\!E}(u,v)_{t_{k}}=\textsf{false}$,

where $\operatorname{\text{\sf\&}}^{\!E}(u,v)_{t_{i}}=u_{s_{i-1}}\wedge v_{s_{0}}$ for all $i$ (since $t_{i}=\operatorname{\sf edge}_{E}(s_{i-1},s_{0})$). Since $v_{s_{0}}=\textsf{true}$, this simplifies to $\operatorname{\text{\sf\&}}^{\!E}(u,v)_{t_{i}}=u_{s_{i-1}}$ which can be directly substituted to (D1) and (D2). Therefore, the conditions for $u$ and $v$ to form an edge are:

1. (1)

$u_{s_{i-1}}\Rightarrow v_{s_{i}}$, for all $i<k$, and 2. (2)

$u_{s_{k-1}}=\textsf{false}$.

This means, in particular, that $u$ has no out-edge (i.e., there is no edge of the form $(u,v)$ for any $v$) if its last entry is true.

It is not hard to see that $\mathbf{L}_{k}$ maps to a dual constructed this way. We can construct a homomorphism $h$ by mapping $i\in\{1,\dots,k\}$ to the tuple starting with $i$ true’s and followed by all false’s, i.e.,

[TABLE]

Note that this is exactly the same homomorphism that is constructed in the proof of Theorem 4.2 (assuming $\mathbf{A}=\mathbf{L}_{k}$). And indeed, it is easy to check that if $i<j$, then $h(i)$ and $h(j)$ satisfy the conditions for an edge given above.

Naturally, there is also a homomorphism the other way. One such homomorphism maps a tuple $u$ that begins with $i$ true’s followed by a false to $i$. Again, it is easy to check that if $(u,v)$ is an edge then $v$ has to begin with one more true. This establishes that our construction is homomorphically equivalent to $\mathbf{L}_{k}$ (as it should be).

Remark 4.4**.**

Let us note that any homomorphism constructed according to the proof of Theorem 4.2 uses only the vertices in the image of $h$ above, i.e., the tuples of the form

[TABLE]

where true appears at least once and false does not need to appear at all. This is quite easy to see: $\mathbf{T}(t_{i})$ maps to $\mathbf{T}(t_{j})$ for all $i<j$ via a homomorphism preserving the roots, hence for any $u$ in the image, we get that if $u_{t_{j}}=\textsf{true}$ then $u_{t_{i}}=\textsf{true}$ for all $i<j$.

We could force similar implications in the definition of the dual by requiring that $u_{t}\Rightarrow u_{s}$ whenever there is a homomorphism $\mathbf{T}(s)\to\mathbf{T}(t)$ preserving roots. We did not include this condition in the definition because our goal is to get a simple construction and not necessarily that the construction results in the smallest graph possible. Nevertheless, this raises a question: Would it be possible by enforcing such implications on our general construction to produce a dual that would be a core (i.e., a structure that is not homomorphically equivalent to any of its proper substructures)?

Finally, we note that our construction of the dual of a tree can be naturally extended to any (finite) tree duality, i.e., given a finite set of finite trees $\mathscr{F}=\{\mathbf{T}_{1},\dots,\mathbf{T}_{n}\}$, we can construct their dual $\mathbf{D}$ that will satisfy that, for any structure $\mathbf{A}$, $\mathbf{A}\to\mathbf{D}$ if and only if for all $i=1,\dots,n$, $\mathbf{T}_{i}\not\to\mathbf{A}$. This $\mathbf{D}$ is constructed as in Definition 4.1 with the following changes: Assume that a term $t_{i}$ represents $\mathbf{T}_{i}$ rooted in an edge for each $i$ (we are assuming that none of $\mathbf{T}_{i}$’s consists of a single vertex). Let $\mathscr{T}$ be a set of terms containing all $t_{i}$’s that is closed under taking subterms, let $\mathscr{T}_{V}$ be the set of all $V$-terms in $\mathscr{T}$, and let $\mathscr{T}_{R}$ be the set of all $R$-terms in $\mathscr{T}$ for each symbol $R$. Finally, replace the condition (D2) with

1. (D2’)

$\operatorname{\text{\sf\&}}^{\!R}(e)_{t_{i}}=\textsf{false}$ for all $i=1,\dots,n$ (naturally, this only applies if $t_{i}$ is an $R$-term).

We denote a structure constructed this way by $\mathbf{D}(t_{1},\dots,t_{n})$.

Theorem 4.5**.**

Let $q_{1},\dots,q_{n}$ be terms for each $i\in\{1,\dots,n\}$, and assume $q_{i}$ is an $R_{i}$-term where $R_{i}$ is a relational symbol. For each structure $\mathbf{A}$ we have that either there exists $i$ such that ${\mathbf{T}(q_{i})}\to\mathbf{A}$, or $\mathbf{A}\to\mathbf{D}(q_{1},\dots,q_{n})$.

Proof 4.6**.**

The proof is essentially identical to the proof of Theorem 4.2. First, the implication ‘${\mathbf{T}(q_{i})}\not\to\mathbf{A}$ for all $i\in\{1,\dots,n\}$ implies $\mathbf{A}\to\mathbf{D}(q_{1},\dots,q_{n})$’ is proven by the same argument as before. In particular, we define $f\colon\mathbf{A}\to\mathbf{D}(q_{1},\dots,q_{n})$ by $f(u)_{t}=\textsf{true}$ if $\mathbf{T}(t)\to\mathbf{A}$ for each $t\in\mathscr{T}_{V}$ and $u\in A$, and $f(u)_{t}=\textsf{false}$ otherwise. The rest of the proof is identical except that instead of arguing that $\operatorname{\text{\sf\&}}^{\!R}(f^{R}(e))$ satisfies (D2) using $\mathbf{Q}\not\to\mathbf{A}$, we argue that $\operatorname{\text{\sf\&}}^{\!R}(f^{R}(e))$ satisfies (D2’) for some $i$ using that ${\mathbf{T}(q_{i})}\not\to\mathbf{A}$.

For the other implication, it is enough to argue that ${\mathbf{T}(q_{i})}\not\to\mathbf{D}(q_{1},\dots,q_{n})$ since then ${\mathbf{T}(q_{i})}$ do not map homomorphically to any $\mathbf{A}$ that maps to $\mathbf{D}(q_{1},\dots,q_{n})$. Observe that Claim 1 and its proof is valid in this case. Consequently, if $h\colon{\mathbf{T}(q_{i})}\to\mathbf{D}(q_{1},\dots,q_{n})$ for some $i$, then $\operatorname{\text{\sf\&}}^{R_{i}}(h^{R_{i}}(r_{q_{i}}))_{q_{i}}=\textsf{true}$ by the claim, which would contradict (D2’).

5. Adjoints to functors not changing the domain

In this section, we describe the simpler of the cases of our construction of an adjoint. We consider $(\sigma,\tau)$-Pultr templates where $\mathbf{P}=\mathbf{V}_{1}$ (i.e., a vertex), and for each $\tau$-symbol $R$, $\mathbf{Q}_{R}$ is a $\sigma$-tree. The homomorphisms $\epsilon_{i,R}\colon\mathbf{P}\to\mathbf{Q}_{R}$ are given by picking elements $x_{1},\dots,x_{\operatorname{ar}R}\in Q_{R}$ that are the images of the unique vertex of $\mathbf{P}$ under $\epsilon_{i,R}$ for the respective $i$’s. Naturally, the elements $x_{1},\dots,x_{\operatorname{ar}R}$ depend on the symbol $R$ which will be always clear from the context. This means that the structure $\Gamma(\mathbf{A})$ can be equivalently described in the following way: the universe of $\Gamma(\mathbf{A})$ coincides with the universe of $\mathbf{A}$, and for every $\tau$-symbol $R$ of arity $k$, we have

[TABLE]

Given such a Pultr template, we will now construct a $\sigma$-structure $\Omega(\mathbf{B})$ from a $\tau$-structure $\mathbf{B}$. In this definition, we use the notation $X\mathbin{\dot{\times}}Y=\{f\cup g\mid f\in X,g\in Y\}$ whenever $X$ and $Y$ are sets of functions defined on disjoint sets (note that there is a natural bijection between $X\mathbin{\dot{\times}}Y$ and $X\times Y$).

Definition 5.1**.**

Fix a $(\sigma,\tau)$-Pultr template with $\mathbf{P}$ being a singleton structure with empty relations and with $\mathbf{Q}_{R}$ being a $\sigma$-tree for each $\tau$-symbol $R$. First, for each $R$, we pick a term $t_{R}$ representing $\mathbf{Q}_{R}$, and define $\mathscr{T}$ to be the set of all subterms of any of the $t_{R}$’s. We use notation $\mathscr{T}_{V}$ and $\mathscr{T}_{S}$ (where $S$ is a $\sigma$-symbol) for the $V$-terms and $S$-terms, respectively, that belong to this set.

The universe of $\Omega(\mathbf{B})$, denoted by $\Omega(B)$, consists of elements

[TABLE]

such that $U_{{\operatorname{\sf vertex}}}$ is a singleton set, i.e., vertices of $\Omega(\mathbf{B})$ are tuples $U$ indexed by $V$-terms in $\mathscr{T}$, where the $t$-th entry is a set of homomorphisms from $\Gamma({\mathbf{T}(t)})$ to $\mathbf{B}$. The ${\operatorname{\sf vertex}}$-th entry $U_{\operatorname{\sf vertex}}$ is then a set containing a single homomorphism which maps $r_{\operatorname{\sf vertex}}$ to some $u\in B$. Note that the relations in $\Gamma({\mathbf{T}(t)})$ will be empty whenever none of the trees $\mathbf{Q}_{R}$ maps to ${\mathbf{T}(t)}$, in which case $U_{t}$ is just some set of functions from the vertices of ${\mathbf{T}(t)}$ to $B$.

For each $\sigma$-symbol $S$ of arity $k$, we define the relation $S^{\Omega(\mathbf{B})}$ as the set of all tuples $E=(U^{1},\dots,U^{k})$ of elements of $\Omega(\mathbf{B})$ for which the tuple $\operatorname{\text{\sf\&}}^{\!S}(E)$, indexed by $\mathscr{T}_{S}$ and defined by

[TABLE]

satisfies:

1. (A1)

For all $s\in\mathscr{T}_{V}$ of the form $s=\operatorname{\sf pr}_{i}(t)$ for some $t\in\mathscr{T}_{S}$, we have $\operatorname{\text{\sf\&}}^{\!S}(E)_{t}\subseteq U^{i}_{s}$. 2. (A2)

For all $t\in\mathscr{T}_{S}$, $\operatorname{\text{\sf\&}}^{\!S}(E)_{t}\subseteq\hom(\Gamma({\mathbf{T}(t)}),\mathbf{B})$.

Let us briefly comment on the definition. First, the definition of $\operatorname{\text{\sf\&}}^{\!S}(E)$, i.e., (3), stems from the observation that in order to define a mapping from the domain of ${\mathbf{T}(t)}$ to $B$, it is enough to specify the values on vertices of ${\mathbf{T}(t_{1})},\dots,{\mathbf{T}(t_{k})}$. The intuition behind condition (A1) is that the actual trees represented by $t$ and $s$ are identical, so any map from the domain of ${\mathbf{T}(t)}$ is a map from the domain of ${\mathbf{T}(s)}$. Finally, the intuition behind condition (A2) is that the union of homomorphisms is a homomorphism (as long as the root edge of $t$ is mapped to an edge).

We claim that this construction $\Omega$ yields a right adjoint to the considered cases of central Pultr functors $\Gamma$.

Theorem 5.2**.**

Assume a $(\sigma,\tau)$-Pultr template with $\mathbf{P}$ being the $\sigma$-structure with a single vertex and empty relations, and $\mathbf{Q}_{R}$ being a $\sigma$-tree for all $\tau$-symbols $R$. Further, assume $\Gamma$ is the central Pultr functor defined by this template, and $\Omega$ is defined as in Definition 5.1.

For every $\sigma$-structure $\mathbf{A}$ and $\tau$-structure $\mathbf{B}$, there is a homomorphism $\Gamma(\mathbf{A})\to\mathbf{B}$ if and only if there is a homomorphism $\mathbf{A}\to\Omega(\mathbf{B})$.

We now proceed to prove the above theorem in several steps. The following lemma proves one of the implications and gives further insights to why $U$’s and $\operatorname{\text{\sf\&}}^{\!S}(E)$’s are defined as above.

Lemma 5.3**.**

If there is a homomorphism $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$, then there is a homomorphism $g\colon\mathbf{A}\to\Omega(\mathbf{B})$.

Proof 5.4**.**

We define a mapping $g\colon A\to\Omega(B)$ by

[TABLE]

We claim that this mapping is a homomorphism. First, we show that $g(u)$ is well-defined, i.e., that $g(u)_{{\operatorname{\sf vertex}}}$ is a singleton set, and the elements of $g(u)_{t}$ are homomorphisms from $\Gamma({\mathbf{T}(t)})$ to $\mathbf{B}$. For the former, observe that $g(u)_{{\operatorname{\sf vertex}}}=\{r_{{\operatorname{\sf vertex}}}\mapsto f(u)\}$ since there is a single homomorphism $h\colon\mathbf{T}({\operatorname{\sf vertex}})\to\mathbf{A}$ which maps the root (and the only vertex) $r_{{\operatorname{\sf vertex}}}$ to $u$. For the latter, assume $t$ is a $V$-term. Since $\mathbf{P}$ is a singleton, it is easy to check that any homomorphism $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ is also a homomorphism $\Gamma({\mathbf{T}(t)})\to\Gamma(\mathbf{A})$. Then $f\circ h\colon\Gamma({\mathbf{T}(t)})\to\mathbf{B}$ is a homomorphism because $f$ is a homomorphism $\Gamma(\mathbf{A})\to\mathbf{B}$.

To prove that $g$ preserves the relations, assume that $e=(u_{1},\dots,u_{k})\in S^{\mathbf{A}}$ is an edge. We first show that

[TABLE]

for each $t\in\mathscr{T}_{S}$, $t=\operatorname{\sf edge}_{S}(t_{1},\dots,t_{k})$. This is true since, for any homomorphism $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ that maps the root edge to $e=(u_{1},\dots,u_{k})$, its restriction $h_{i}\colon{\mathbf{T}(t_{i})}\to\mathbf{A}$ maps the roots to the respective $u_{i}$ for all $i$, and, for any tuple of homomorphisms $h_{i}\colon{\mathbf{T}(t_{i})}\to\mathbf{A}$ which maps the roots to the respective $u_{i}$’s, their union is a homomorphism ${\mathbf{T}(t)}\to\mathbf{A}$.

To prove property (A1), we need to check that $\operatorname{\text{\sf\&}}^{\!S}(g^{S}(e))_{t}\subseteq g(u_{i})_{\operatorname{\sf pr}_{i}(t)}$. This is easy to see since any homomorphism $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ that maps $r_{t}$ to $e$ maps $r_{\operatorname{\sf pr}_{i}(t)}$, which is the $i$-th component of $r_{t}$, to $u_{i}$. Finally, the property (A2), that each mapping in $\operatorname{\text{\sf\&}}^{\!S}(g^{S}(e))_{t}$ is a homomorphism, is proved in the same way as the analogous statement for $g(u)$, i.e., it follows from the above claim.

The above lemma concludes one of the implications that we need for the adjunction. We turn to the other implication which we prove in two steps, each provided by one of the following two lemmas. The first lemma assures the adjunction in a special case when $\mathbf{A}={\mathbf{T}(t)}$ for some $t\in\mathscr{T}$. This will be used in the proof of the general case.

Lemma 5.5**.**

Let $t\in\mathscr{T}$ and let $h\colon{\mathbf{T}(t)}\to\Omega(\mathbf{B})$ be a homomorphism, and $d\colon T(t)\to B$ is the mapping such that $d(v)$ is the value of the single map in $h(v)_{{\operatorname{\sf vertex}}}$ for all vertices $v$ of ${\mathbf{T}(t)}$. Then $d$ is a homomorphism $\Gamma({\mathbf{T}(t)})\to\mathbf{B}$.

Proof 5.6**.**

We prove by induction on $t$ that $d\in h(r_{t})_{t}$ if $t$ is a $V$-term and $d\in\operatorname{\text{\sf\&}}^{\!S}(h^{S}(r_{t}))_{t}$, if $t$ is an $S$-term.

**Case $t={\operatorname{\sf vertex}}$.: **

This is a trivial case.

**Case $t=\operatorname{\sf edge}_{S}(t_{1},\dots,t_{k})$.: **

Note that restrictions of $h$ to subtrees ${\mathbf{T}(t_{i})}$ are homomorphisms, so we know that, for all $i$, $h(r_{t_{i}})_{t_{i}}$ contains the restrictions of $d$ by the inductive assumption. The claim then immediately follows from the definition of $\operatorname{\text{\sf\&}}^{\!S}(h^{S}(r_{t}))_{t}$.

**Case $t=\operatorname{\sf pr}_{i}(t^{\prime})$.: **

Since $h$ is a homomorphism from ${\mathbf{T}(t)}={\mathbf{T}(t^{\prime})}$ to $\mathbf{B}$, we know that $\operatorname{\text{\sf\&}}^{\!S}(h^{S}(r_{t^{\prime}}))_{t^{\prime}}$ contains $d$, and the claim subsequently follows by (A1).

The lemma then immediately follows either by the definition, if $t$ is a $V$-term, or by (A2), otherwise.

Note that the induction in the above proof alternates between $V$-terms and $S$-terms. We use the above lemma to prove the general case.

Lemma 5.7**.**

If $g\colon\mathbf{A}\to\Omega(\mathbf{B})$ is a homomorphism, then there is a homomorphism $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$.

Proof 5.8**.**

We assume that $\mathbf{Q}_{R}={\mathbf{T}(t_{R})}$ — this assumption is without loss of generality, since by definition $\mathbf{Q}_{R}$ and ${\mathbf{T}(t_{R})}$ are isomorphic. We define $f$ by setting $f(u)$ to be the unique value attained by the single map in $g(u)_{{\operatorname{\sf vertex}}}$. This is a well-defined mapping on the vertices of $\Gamma(\mathbf{A})$. We need to show that it preserves the relations of $\Gamma(\mathbf{A})$. To this end, assume that $R$ is a $\tau$-symbol of arity $k$ and $(u_{1},\dots,u_{k})\in R^{\Gamma(\mathbf{A})}$. This means that there is a homomorphism $h\colon\mathbf{Q}_{R}\to\mathbf{A},$ s.t., $h(x_{i})=u_{i}$ for all $i\in[k]$. Observe that $g\circ h\colon{\mathbf{T}(t_{R})}\to\Omega(\mathbf{B})$ is a homomorphism since it is obtained as a composition of two homomorphisms, so Lemma 5.5 applies to $g\circ h$ in place of $h$ and $f\circ h$ in place of $d$ (since $fh(v)$ is the unique value attained by the single map in $gh(v)_{{\operatorname{\sf vertex}}}$). Consequently, $f\circ h$ is a homomorphism from $\Gamma({\mathbf{T}(t_{R})})$ to $\mathbf{B}$, and therefore

[TABLE]

since $h(x_{i})=u_{i}$ and $(x_{1},\dots,x_{k})\in R^{\Gamma({\mathbf{T}(t_{R})})}$, where the latter follows from the definition of $R^{\Gamma({\mathbf{T}(t_{R})})}$ as phrased in (2) by letting $r$ be the identity homomorphism on ${\mathbf{T}(t_{R})}$.

Lemmas 5.3 and 5.7 together yield Theorem 5.2.

Remark 5.9**.**

Let us briefly compare our general construction to that of Foniok and Tardif [FT15, Theorem 7.1] assuming that both signatures are digraphs and $\mathbf{P}=\mathbf{V}_{1}$ (denoted by $\vec{P}_{0}$ in [FT15]). An analogous comparison also applies for the construction presented in the next section. Also, examples below compare the two constructions in several concrete cases of central Pultr functors.

First, both the domain of our right adjoint $\Omega$ and Foniok-Tardif right adjoint $\Omega_{\text{FT}}$ consist of some tuples indexed by some rooted subtrees of $\mathbf{Q}_{E}$. First difference is how do we choose the subtrees. Foniok and Tardif choose a vertex of $\mathbf{Q}_{E}$ called middle vertex which disconnects the two copies of $\mathbf{P}$ in $\mathbf{Q}_{E}$, and define the set of subtrees using this vertex. This is similar to choosing a term representing $\mathbf{Q}_{E}$ rooted in the middle vertex, although not every subtree used by Foniok and Tardif needs to be represented by a subterm. Furthermore, note that the requirement that the middle vertex separates the two copies of $\mathbf{P}$ creates an obstacle to generalising Foniok and Tardif’s construction to relational structures with higher arity.

Second, the elements of tuples composing the universe of $\Omega_{\text{FT}}$ are subsets of vertices of $\mathbf{B}$ while elements of our tuples are sets of mappings to $B$. The main reason that Foniok and Tardif are able to do this is that the middle vertex separates the two copies of $\mathbf{P}$, and hence they only need to track the value of the mappings on one distinguished vertex. Again, since our aim is to provide adjoints in a more general setting, we cannot afford to do that.

5.1. Example: An oriented path

In this example, we compare our construction to the construction introduced in [FT15, Definition 4.1]. Our goal is to construct the adjoint to the digraph Pultr functor $\Gamma$ defined by the Pultr template where $\mathbf{Q}_{E}$ is the following digraph:

[TABLE]

The maps $\epsilon_{1,E}$ and $\epsilon_{2,E}$ map the singleton $\mathbf{P}$ to $x_{1}$ and $x_{2}$, respectively.

Let us start by fixing a term $t_{E}$ representing $\mathbf{Q}_{E}$. Namely, we let

[TABLE]

which represents $\mathbf{Q}_{E}$ rooted in the middle edge. It has two $V$-subterms and two $E$-subterms (including itself) that represent the following trees:

[TABLE]

For a directed graph $\mathbf{H}$, the definition of $\Omega(\mathbf{H})$ is spelled out as follows. The vertices of $\Omega(\mathbf{H})$ are pairs $U=(U_{s_{0}},U_{s_{1}})$ where $U_{s_{0}}$ is the set containing the map that sends the unique vertex of ${\mathbf{T}({\operatorname{\sf vertex}})}$ to some $u_{0}\in H$ and $U_{s_{1}}\subseteq H^{T(s_{1})}$; this is because $\Gamma(\mathbf{T}(s_{1}))$ has no edges since $\mathbf{Q}_{E}\not\to\mathbf{T}(s_{1})$. There is an edge from $U=(U_{s_{0}},U_{s_{1}})$ to $V=(V_{s_{0}},V_{s_{1}})$ if the tuple $\operatorname{\text{\sf\&}}^{\!E}(U,V)$ defined by

[TABLE]

satisfies

1. (1)

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}\subseteq U_{s_{1}}$, and 2. (2)

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}\subseteq\hom(\Gamma(\mathbf{Q}_{E}),\mathbf{H})$

Let us simplify this definition. First, we will write homomorphisms from the above paths as tuples, writing the values of such homomorphisms from left to right as the vertices appear on the picture above. In this way, we have $U_{s_{0}}\subseteq H$, $U_{s_{1}}\subseteq H\times H$ for each $U$, and $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}=U_{s_{0}}\times V_{s_{0}}$,

[TABLE]

The two conditions are then rephrased as follows:

1. (1)

$U_{s_{0}}\times V_{s_{0}}\subseteq U_{s_{1}}$, and 2. (2)

$(u_{1},v_{1})\in E^{\mathbf{H}}$ for every $(u_{1},u_{0},v_{0},v_{1})\in\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}$, i.e., for every $(u_{0},u_{1})\in U_{s_{1}}$ and $(v_{0},v_{1})\in V_{s_{1}}$.

We claim that this construction results (on the same input) in a digraph that is homomorphically equivalent to the one obtained by [FT15, Definition 4.1]. Namely, the adjoint constructed there, let us call it $\Omega^{\prime}$, is as follows: The vertices of $\Omega^{\prime}(\mathbf{H})$ are pairs $(a,A)$ where $a\in H$ and $A\subseteq H$, and there is an edge from $(a,A)$ to $(b,B)$ if $b\in A$ and $A\times B\subseteq E^{\mathbf{H}}$.

We show that, for every graph $\mathbf{H}$, there is a homomorphism $\alpha\colon\Omega^{\prime}(\mathbf{H})\to\Omega(\mathbf{H})$ defined by $\alpha((a,A))=U$ where $U_{s_{0}}=\{a\}$ and $U_{s_{1}}=\{a\}\times A$. To show that $\alpha$ preserves edges, assume that $(a,A)$ and $(b,B)$ are connected by an edge in $\Omega^{\prime}(\mathbf{H})$, i.e., $b\in A$ and $A\times B\subseteq E^{\mathbf{H}}$, and $U=\alpha((a,A))$, $V=\alpha((b,B))$. We have that

[TABLE]

and claim that:

• •

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}\subseteq U_{s_{1}}$. This is true since $b\in A$.

• •

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}\subseteq\hom(\Gamma(\mathbf{Q}_{E}),\mathbf{H})$. This is true, since the only edge of $\Gamma(\mathbf{Q}_{E})$ is $(x_{1},x_{2})$ and the projection of $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}$ to $x_{1},x_{2}$ (the first and the last coordinate) is $A\times B\subseteq E^{\mathbf{H}}$.

A homomorphism $\beta\colon\Omega(\mathbf{H})\to\Omega^{\prime}(\mathbf{H})$ is given by $\beta(U)=(a,A)$ where $a$ is the unique element of $U_{s_{0}}$, and

[TABLE]

To show that it is a homomorphism, assume $(U,V)\in E^{\Omega(\mathbf{H})}$, and let $\beta(U)=(a,A)$ and $\beta(V)=(b,B)$. Since $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}=U_{s_{0}}\times V_{s_{0}}=\{(a,b)\}$ and $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}\subseteq U_{s_{1}}$, we have $(a,b)\in U_{s_{1}}$, which implies that $b\in A$. Also since $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}\subseteq\hom(\Gamma(\mathbf{Q}_{E}),\mathbf{H})$, $a\times A\subseteq U_{s_{1}}$, $b\times B\subseteq V_{s_{1}}$, and $t_{E}=\operatorname{\sf edge}_{E}(s_{1},s_{1})$, we have that

[TABLE]

where the first inclusion follows from the definition of $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}$, and the second includion follows from (A2). In particular, the inclusion above implies that $A\times B\subseteq E^{\mathbf{H}}$. This concludes the proof of the homomorphic equivalence of $\Omega(\mathbf{H})$ and $\Omega^{\prime}(\mathbf{H})$.

We note that our $\Omega(\mathbf{H})$ can be reduced to a smaller homomorphically equivalent structure by requiring that vertices $U\in\Omega(\mathbf{H})$ satisfy

1. (A3)

for all $s,s^{\prime}\in\mathscr{T}_{V}$ and homomorphisms $h\colon{\mathbf{T}(s)}\to\mathbf{T}(s^{\prime})$ with $h(r_{s})=r_{s^{\prime}}$, we have

[TABLE]

Note that the elements constructed in the proof of Lemma 5.3 satisfy this property. In this particular example, this requirement would force that $U_{s_{1}}=a\times A$ for some $A\subseteq H$ and $a\in U_{s_{0}}$ since $s_{0}$ is embedded to $s_{1}$ as the root. This would then make the two homomorphisms defined above isomorphisms.

5.2. Example: A 4-ary relation defined by an oriented path

Our definition works also for Pultr templates that are not just digraph templates. As an example for comparison, let us consider a Pultr template that is similar to the previous example, but in this case maps digraphs to structures over a signature containing one 4-ary relational symbol $R$.

Specifically, $\mathbf{P}$ is still a singleton with no edges, and $\mathbf{Q}_{R}$ is the same digraph as $\mathbf{Q}_{E}$ above, but we now have 4 homomorphisms $\epsilon_{i,R}\colon\mathbf{P}\to\mathbf{Q}_{R}$ for $i=0,1,2,3$ which map the vertex of $\mathbf{P}$ to $0,1,2$, or $3$ respectively. Pictorially, the digraph $\mathbf{Q}_{R}$ together with its distinguished vertices $x_{0}$, …, $x_{3}$ is as follows.

[TABLE]

We let $t_{R}=t_{E}$ where $t_{E}$ is as above, and we use the same notation as in the previous example.

For a structure $\mathbf{B}$ with a 4-ary relation $R^{\mathbf{B}}$, vertices of $\Omega(\mathbf{B})$ are defined in a similar way as above, i.e., they are pairs $(U_{s_{0}},U_{s_{1}})$ where $U_{s_{0}}=\{b\}$ for some $b\in B$ and $U_{s_{1}}\subseteq B\times B$. Two such vertices $(U,V)$ are connected by an edge if $\operatorname{\text{\sf\&}}^{\!E}(U,V)$ defined by $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{R}}=U_{s_{1}}\mathbin{\dot{\times}}V_{s_{1}}$ and $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}=U_{s_{0}}\mathbin{\dot{\times}}V_{s_{0}}$ satisfies

1. (1)

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{R}}\subseteq R^{\mathbf{B}}$, and 2. (2)

$\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{1}}\subseteq U_{s_{1}}$.

More precisely, the first condition can be written as

[TABLE]

and the second condition can be written as $U_{s_{0}}\times V_{s_{0}}\subseteq U_{s_{1}}$.

The only real difference from the above example is that $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{R}}\subseteq R^{\mathbf{B}}$ instead of requiring that the projection of $\operatorname{\text{\sf\&}}^{\!E}(U,V)_{t_{E}}$ on the first and the last coordinates is a subset of $E^{\mathbf{H}}$. As above, we could create a homomorphically equivalent $\Omega^{\prime}(\mathbf{B})$ whose vertices would be pairs $(u,U)$ where $u\in B$ and $U\subseteq B$. Two such vertices $(u,U)$ and $(v,V)$ would be then connected by an edge if $v\in U$ and

[TABLE]

5.3. Duals from adjoints

In [FT15], the authors claim that for digraph Pultr templates with the edge relation defined by $\mathbf{Q}_{E}$, the image of the digraph with a single vertex and no edges under the right Pultr functor is a dual to $\mathbf{Q}_{E}$. We may be slightly more precise when talking about our constructions, namely, we claim the following.

Proposition 5.10**.**

Let $\tau$ contain a single symbol $R$, and fix a $(\sigma,\tau)$-Pultr template defined by $\mathbf{P}=\mathbf{V}_{1}$, and $\mathbf{Q}_{R}={\mathbf{T}(t_{R})}$ for some term $t_{R}$ such that $\mathbf{Q}_{R}\not\to{\mathbf{T}(t)}$ for any $t<t_{R}$ (i.e., $t_{R}$ is a minimal term among those representing a structure homomorphically equivalent to $\mathbf{Q}_{R}$). Let $\Omega$ be the right adjoint to $\Gamma$ as defined in Definition 5.1. Then the image of the $\tau$-structure $\mathbf{V}_{1}$ under $\Omega$ is isomorphic to $\mathbf{D}(t_{R})$.

Proof 5.11**.**

Observe that there is only one function from any set $T$ to $V_{1}=\{1\}$. Hence, there is only one candidate function $f$ for a homomorphism $\Gamma({\mathbf{T}(t)})\to\mathbf{V}_{1}$ for any term $t$. It is not hard to observe that $f$ is a homomorphism if, and only if, $\Gamma({\mathbf{T}(t)})$ has no edges. In particular, $f$ is a homomorphism for all $t<t_{r}$, since $\mathbf{Q}_{R}\not\to{\mathbf{T}(t)}$ in that case. Consequently, each of the components of an element $U\in\Omega(V_{1})$ is either the empty set or a singleton set. The proof relies on the fact that if we interpret $\emptyset$ as false and a non-empty set as true, then $X\subseteq Y$ is equivalent to $X\Rightarrow Y$, and $X\times Y$ is equivalent to $X\wedge Y$.

We define a map $h\colon\Omega(V_{1})\to D(t_{R})$, where $D(t_{R})$ is the domain of $\mathbf{D}(t_{R})$, by $h(U)=u$ where

[TABLE]

Observe that a similar correspondence holds also for $\operatorname{\text{\sf\&}}^{\!R}(e)$ and $\operatorname{\text{\sf\&}}^{\!R}(E)$ where $E\in R^{\Omega(V_{1})}$ and $h^{R}(E)=e$, i.e., $\operatorname{\text{\sf\&}}^{\!R}(e)_{t}=\textsf{true}$ if $\operatorname{\text{\sf\&}}^{\!R}(E)_{t}\neq\emptyset$, and $\operatorname{\text{\sf\&}}^{\!R}(e)_{t}=\textsf{false}$ if $\operatorname{\text{\sf\&}}^{\!R}(E)_{t}=\emptyset$, since the product of some sets is non-empty if and only if all of the sets are non-empty. We get that (A1) implies (D1) since a subset of the empty set is empty. And, (D2) is satisfied since $\Gamma({\mathbf{T}(t_{R})})$ has an edge (witnessed by the identity homomorphism), and therefore there is no homomorphism from this structure to $\mathbf{V}_{1}$, and consequently $\operatorname{\text{\sf\&}}^{\!R}(E)_{t_{R}}=\emptyset$ by (A2) and $e_{t_{R}}=\textsf{false}$. This concludes that $h$ is a well-defined homomorphism.

Finally, observe that $h$ is an invertible map since there is a unique non-empty subset of the singleton set $\hom(\Gamma({\mathbf{T}(t)}),\mathbf{V}_{1})$ for all $t<t_{R}$. The inverse is again a homomorphism since all the implications we argued above can be reversed, i.e., if $e=(u^{1},\dots,u^{k})\in R^{\mathbf{D}(t_{R})}$, and $E$ is such that $h(E)=e$, then $\operatorname{\text{\sf\&}}^{\!R}(E)_{t}=\emptyset$ if and only if $\operatorname{\text{\sf\&}}^{\!R}(e)_{t}=\textsf{false}$. The conditions (A1) and (A2) then follow from (D1) and (D2), respectively.

6. Adjoints to functors with domains defined by a relation

In this section, we construct a right adjoint to the central Pultr functor defined by a $(\sigma,\tau)$-Pultr template for which $\mathbf{P}={\mathbf{\check{S}}}_{1}$, for some choice of $\sigma$-symbol $\check{S}$, i.e., $\mathbf{P}$ is an $\check{S}$-edge. Again, for each $k$-ary $\tau$-symbol $R$, we have a $\sigma$-tree $\mathbf{Q}_{R}$ with a $k$-tuple $\epsilon_{1,R}$, …, $\epsilon_{k,R}$ of homomorphisms $\mathbf{P}\to\mathbf{Q}_{R}$. Each of these homomorphisms selects an $\check{S}$-edge of $\mathbf{Q}_{R}$, we denote these edges by $x_{1}$, …, $x_{k}$, respectively, i.e., $x_{i}=\epsilon_{i,R}^{\check{S}}(e)$ for each $i=1,\dots,k$ where $e\in\check{S}^{\mathbf{P}}$ is the unique $\check{S}$-edge of $\mathbf{P}$. Naturally, the edges $x_{1},\dots,x_{\operatorname{ar}R}$ depend on the symbol $R$ which will be always clear from the context. Let us repeat the definition of $\Gamma(\mathbf{A})$ for a $\sigma$-structure $\mathbf{A}$ using this notation: The domain of $\Gamma(\mathbf{A})$ is $\check{S}^{\mathbf{A}}$, and for each $\tau$-symbol $R$ of arity $k$, the corresponding relation of $\Gamma(\mathbf{A})$ is defined as

[TABLE]

We also note that, for each homomorphism $h\colon\mathbf{A}\to\mathbf{B}$ between two $\sigma$-structures $\mathbf{A}$ and $\mathbf{B}$, $h^{\check{S}}\colon\Gamma(\mathbf{A})\to\Gamma(\mathbf{B})$ is a homomorphism.

The construction of $\Omega$ in this case is very similar to the construction in Section 5.2 (Definition 5.1), with a few small changes.

Definition 6.1**.**

Fix a $(\sigma,\tau)$-Pultr template with $\mathbf{P}={\mathbf{\check{S}}}_{1}$ for some $\sigma$-symbol $\check{S}$, and assume $\mathbf{Q}_{R}$ is a $\sigma$-tree for each $\tau$-symbol $R$. For each $\tau$-symbol $R$, we pick a term $t_{R}$ representing $\mathbf{Q}_{R}$, and we let $\mathscr{T}$ be the set of all subterms of any of the $t_{R}$’s. We use notation $\mathscr{T}_{V}$ and $\mathscr{T}_{S}$ for the $V$-terms and $S$-terms, where $S$ is a $\sigma$-symbol, that belong to this set. Now, we are ready to define $\Omega(\mathbf{B})$.

The universe of $\Omega(\mathbf{B})$ consists of elements

[TABLE]

such that $U_{{\operatorname{\sf vertex}}}=\{\emptyset\}$ (here $\emptyset$ denotes the empty map; note that $\Gamma({\mathbf{T}({\operatorname{\sf vertex}})})$ has no vertices). We impose this trivial definition of the component $U_{\operatorname{\sf vertex}}$ to avoid complicated case distinction below.

For each $\sigma$-symbol $S$ of arity $k$, different from $\check{S}$, we define the relation $S^{\Omega(\mathbf{B})}$ to be the set of all tuples $E=(U^{1},\dots,U^{k})$ for which the tuple $\operatorname{\text{\sf\&}}^{\!S}(E)$, indexed by $\mathscr{T}_{S}$ and defined by

[TABLE]

(the expression makes sense because $\check{S}^{{\mathbf{T}(t)}}=\check{S}^{\mathbf{T}(t_{1})}\cup\dots\cup\check{S}^{\mathbf{T}(t_{k})}$, satisfies:

1. (B1)

For all $s\in\mathscr{T}_{V}$ of the form $s=\operatorname{\sf pr}_{i}(t)$ for some $t\in\mathscr{T}_{S}$, we have $\operatorname{\text{\sf\&}}^{\!S}(E)_{t}\subseteq U^{i}_{s}$. 2. (B2)

For all $t\in\mathscr{T}_{S}$, $\operatorname{\text{\sf\&}}^{\!S}(E)_{t}\subseteq\hom(\Gamma({\mathbf{T}(t)}),\mathbf{B})$.

(Up to this point, this definition is almost identical to Definition 5.1).

Finally, the relation $\check{S}^{\Omega(\mathbf{B})}$ is defined to be the set of all tuples $E=(U^{1},\dots,U^{k})$, where $k=\operatorname{ar}\check{S}$, for which there exist $e_{\bullet}\in B$ such that the tuple $\operatorname{\text{\sf\&}}^{\!\check{S}}(E,e_{\bullet})$, indexed by elements $t=\operatorname{\sf edge}_{\check{S}}(t_{1},\dots,t_{k})$ of $\mathscr{T}_{\check{S}}$ and defined by

[TABLE]

(the expression makes sense because $\check{S}^{{\mathbf{T}(t)}}=\check{S}^{\mathbf{T}(t_{1})}\cup\dots\cup\check{S}^{\mathbf{T}(t_{k})}\cup\{r_{t}\}$), satisfies (B1) and (B2) with $\check{S}$ in place of $S$ and $\operatorname{\text{\sf\&}}^{\!\check{S}}(E,e_{\bullet})$ in place of $\operatorname{\text{\sf\&}}^{\!S}(E)$.

We call the element $e_{\bullet}$, for which $\operatorname{\text{\sf\&}}^{\!\check{S}}(E,e_{\bullet})$ satisfies the conditions (B1) and (B2) for the $\check{S}$-edge $E=(U_{1},\dots,U_{k})$, a witness of this edge. Also if $E\in\check{S}^{\Omega(\mathbf{B})}$ and $e_{\bullet}$ is a (potential) witness, we will write $\operatorname{\text{\sf\&}}^{\!\check{S}}(E,e_{\bullet})$ for the tuple defined by (4).

The reason why (4) differs from the previous definition is that the domain of $\Gamma(\mathbf{T}(t))$ contains the element $r_{t}$, which does not belong to any of the domains of $\Gamma(\mathbf{T}(t_{i}))$’s, unlike in the previous cases. This means that in order to define a homomorphism from $\Gamma(\mathbf{T}(t))$, we need to specify not only values at vertices of $\Gamma(\mathbf{T}(t_{i}))$’s but also at $r_{t}$. This explains the necessity for the introduction of this new element $e_{\bullet}$.

We claim that this definition indeed constructs a right adjoint to $\Gamma$. The proof is very similar to the proof of Theorem 5.2, and we leave it for the last subsection of this section.

Theorem 6.2**.**

Assume a $(\sigma,\tau)$-Pultr template with $\mathbf{P}$ being the $\sigma$-tree with $\operatorname{ar}\check{S}$ vertices connected by an $\check{S}$-edge for some $\sigma$-symbol $\check{S}$, and $\mathbf{Q}_{R}$ being a $\sigma$-tree for all $\tau$-symbols $R$. Further assume $\Gamma$ is the central Pultr functor defined by this template, and $\Omega$ is defined as in Definition 6.1.

For every $\sigma$-structure $\mathbf{A}$ and $\tau$-structure $\mathbf{B}$, there is a homomorphism $\Gamma(\mathbf{A})\to\mathbf{B}$ if and only if there is a homomorphism $\mathbf{A}\to\Omega(\mathbf{B})$.

6.1. Example: The arc graph construction

Recall the arc-graph construction from Example 2.8 which can be expressed as the central Pultr functor whose template consists of structures $\mathbf{P}=(\{0,1\};\{(0,1)\})$ and $\mathbf{Q}_{E}=(\{0,1,2\};\{(0,1),(1,2)\})$ with $\epsilon_{1}(0)=0$ and $\epsilon_{1}(1)=1$, and $\epsilon_{2}(0)=1$ and $\epsilon_{2}(1)=2$.

The right adjoint $\Omega$ according to our definition above would be constructed in the following way: First, we choose a term $t_{E}$ representing $\mathbf{Q}_{E}$. We can pick $t_{2}$ as in Section 4.1.

[TABLE]

We also name all its subterms as in Section 4.1, i.e.,

[TABLE]

Vertices of $\Omega(\mathbf{B})$ are defined as pairs $(U_{s_{0}},U_{s_{1}})$ such that $U_{s_{0}}=\{\emptyset\}$ and $U_{s_{1}}\subseteq B^{T(s_{1})}$ (note that ${\mathbf{T}(s_{1})}$ has a single vertex and no edges). Two such vertices $U,V$ are connected by an edge if there is $e_{\bullet}\in B$ such that $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})$, defined by

[TABLE]

satisfies

1. (1)

$\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{1}}\subseteq V_{s_{1}}$ — this is condition (B1) for $s_{1}=\operatorname{\sf pr}_{2}(t_{1})$; 2. (2)

$\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{i}}\subseteq\hom(\Gamma(\mathbf{T}(t_{i})),\mathbf{B})$ for $i=1,2$ — this is condition (B2).

These conditions can be considerably simplified. First, since $U_{s_{0}}=V_{s_{0}}=\{\emptyset\}$, both can be dropped from the above products, and also they carry no additional information. Further, since $\Gamma(\mathbf{T}(s_{1}))=\Gamma(\mathbf{T}(t_{1}))$ is the graph with a single vertex and no edges, we can identify $U_{s_{1}}$ and $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{1}}$ with subsets of $B$. Connecting this observation with the comment about $U_{s_{0}}$, we can identify $U$ with $U_{s_{1}}\subseteq B$. Finally, the elements of $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{2}}$ are homomorphisms from a directed edge to $\mathbf{B}$ which correspond to the edges of $\mathbf{B}$. Using this correspondence, we can identify $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{2}}$ with a subset of $E^{\mathbf{B}}$. Taking all of these into account the conditions above simplify to

1. (1)

$\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{1}}=\{e_{\bullet}\}\subseteq V_{s_{1}}$, and 2. (2)

$\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{2}}=U_{s_{1}}\times\{e_{\bullet}\}\subseteq E^{\mathbf{B}}$.

So, we can say vertices of $\Omega(\mathbf{B})$ are subsets $U$ of $B$, and $(U,V)$ is an edge of $\Omega(\mathbf{B})$ if

[TABLE]

We compare this construction with the functor $\delta_{R}$ described in [FT15, Definition 3.1] as a right adjoint to $\delta$. For a digraph $\mathbf{B}$, the vertices of the digraph $\delta_{R}(\mathbf{B})$ are the complete bipartite subgraphs of $\mathbf{B}$, i.e., pairs $(U^{-},U^{+})$ of subsets of vertices of $\mathbf{B}$ such that $U^{-}\times U^{+}\subseteq E^{\mathbf{B}}$. There is an edge from $(U^{-},U^{+})$ to $(V^{-},V^{+})$ if $U^{+}\cap V^{-}\neq\emptyset$. Below, we show that $\delta_{R}(\mathbf{B})$ and $\Omega(\mathbf{B})$ are homomorphically equivalent.

We start with constructing a homomorphism $h\colon\delta_{R}(\mathbf{B})\to\Omega(\mathbf{B})$. We let

[TABLE]

To show that it preserves edges, assume $U^{+}\cap V^{-}\neq\emptyset$, i.e., there exists $e_{\bullet}\in U^{+}\cap V^{-}$. We claim that this $e_{\bullet}$ witnesses that $U^{-}$ and $V^{-}$ is an edge in $\Omega(\mathbf{B})$. Clearly, $e_{\bullet}\in V^{-}$. Also, we have

[TABLE]

A homomorphism $g\colon\Omega(\mathbf{B})\to\delta_{R}(\mathbf{B})$ is a bit harder to construct. Guided by the above, it is natural to choose the first component of $g(U)$ to be $g(U)^{-}=U$. We need to define the second component $g(U)^{+}$. We let $g(U)^{+}$ be the largest set such that $g(U)^{-}\times g(U)^{+}\subseteq E^{\mathbf{B}}$, i.e.,

[TABLE]

Now, assume that $U$ and $V$ are connected by an edge in $\Omega(\mathbf{B})$ witnessed by $e_{\bullet}$. We claim that $e_{\bullet}\in g(U)^{+}\cap g(V)^{-}$. By definition of $\Omega(\mathbf{B})$, we have $e_{\bullet}\in V=g(V)^{-}$, and $U\times\{e_{\bullet}\}\subseteq E^{\mathbf{B}}$, which implies that $e_{\bullet}\in g(U)^{+}$. Altogether, $e_{\bullet}\in g(U)^{+}\cap g(V)^{-}$, and hence $(g(U),g(V))\in E^{\delta_{R}(\mathbf{B})}$. This completes the proof.

This example shows how $U_{{\operatorname{\sf vertex}}}$ can be eliminated from the definition of $\Omega(\mathbf{B})$. Let us repeat again, that the only purpose of $U_{{\operatorname{\sf vertex}}}$ is to avoid case distinction between some $E$-terms, e.g., between $\operatorname{\sf edge}_{E}(t,s)$ and $\operatorname{\sf edge}_{E}(t,{\operatorname{\sf vertex}})$ for $s\neq{\operatorname{\sf vertex}}$. We may ignore it in this example since we expanded every single case when it is used.

6.2. Example: Arc structure

In this subsection, we consider a certain variant of the arc graph construction which we will call an arc structure and which encodes more information than the arc graph: The domain of the arc structure coincides with the domain of the arc graph, i.e., the set of all edges of the input graph, and we extend the signature with two more binary symbols that will relate those pairs of edges that are incident in a different sense. The goal of this example is two-fold: first, to show how the construction of right adjoint works in a more general signature, and second, to show how the right adjoint changes if we change the central Pultr functor in such a way that it encodes more information about the input structure. Note, for example, that any tree can be recovered from its image under the arc structure construction, but there are trees (e.g., the one presented in Fig. 1) that cannot be recovered from their arc graph.

We fix $\phi$ to be a signature with three binary relations $D$, $I$, and $O$, and we let $\gamma$ be the signature of digraphs. We define a central Pultr functor $\partial$ using the $(\gamma,\phi)$-Pultr template defined as follows: The digraph defining vertices is the digraph with a single directed edge, i.e., $\mathbf{P}=\mathbf{E}_{1}$, and the digraphs $\mathbf{Q}_{D}$, $\mathbf{Q}_{I}$, $\mathbf{Q}_{O}$ are the following where the images of $\epsilon_{i}$’s are highlighted and labelled by $x_{i}$.

[TABLE]

Note that the digraph $\mathbf{Q}_{D}$ with the two distinguished edges defines the arc-graph functor. This means that for each digraph $\mathbf{G}$, the reduct $(A,D^{\mathbf{A}})$ where $\mathbf{A}=\partial(\mathbf{G})$ is the arc-graph of $\mathbf{G}$ (if $D$ is interpreted as $E$), see Figure 1.

Now, to construct the right adjoint to $\partial$, which we denote by $\omega_{\partial}$, we fix the following terms representing the graphs $\mathbf{Q}_{D}$, $\mathbf{Q}_{I}$, and $\mathbf{Q}_{O}$.

[TABLE]

where $s_{i}=\operatorname{\sf pr}_{i}(\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}}))$ for $i=1,2$. We name all remaining subterms as follows, $s_{0}={\operatorname{\sf vertex}}$, $t_{E}=\operatorname{\sf edge}_{E}({\operatorname{\sf vertex}},{\operatorname{\sf vertex}})$. The set of terms defining $\omega_{\partial}$ is $\mathscr{T}=\{s_{0},s_{1},s_{2},t_{E},t_{D},t_{I},t_{O}\}$. Of which, the $V$-terms are $\mathscr{T}_{V}=\{s_{0},s_{1},s_{2}\}$. Which means that the vertices of $\omega_{\partial}(\mathbf{B})$ are triples $(U_{s_{0}},U_{s_{1}},U_{s_{2}})$ where $U_{s_{0}}=\{\emptyset\}$, and $U_{s_{1}},U_{s_{2}}$ are sets of functions from a 1-element set to $B$. We identify such a triple with a pair $(U^{+},U^{-})$ where $U^{+}\subseteq B$ is the set of images of function in $U_{s_{1}}$ and $U^{-}\subseteq B$ the set of images of functions in $U_{s_{2}}$. Using a similar simplifications of the definition of edges as in the previous example, we get that two such pairs $(U^{+},U^{-})$ and $(V^{+},V^{-})$ are connected by an edge if there exists $e_{\bullet}\in B$, so that the sets $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{E}}=\{e_{\bullet}\}$, $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{D}}=U^{-}\times\{e_{\bullet}\}$, $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{I}}=\{e_{\bullet}\}\times V^{-}$, and $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{O}}=U^{+}\times\{e_{\bullet}\}$ satisfy $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{E}}\subseteq U^{+}\cap V^{-}$ and $\operatorname{\text{\sf\&}}^{\!E}((U,V),e_{\bullet})_{t_{R}}\subseteq R^{\mathbf{B}}$ for $R\in\{D,I,O\}$. This means that $(U^{+},U^{-})$ and $(V^{+},V^{-})$ are connected by an edge if there exists $e_{\bullet}\in U^{+}\cap V^{-}$ such that

[TABLE]

This completes the definition, though we can further refine it by requiring that for each vertex, the sets $U^{+}$ and $U^{-}$ satisfy additional properties that would automatically imply the conditions above. Namely, we require

[TABLE]

To sum up the refined definition, we let $\omega_{\partial}(\mathbf{B})$ be the digraph with vertex set

[TABLE]

where $(U^{+},U^{-})$ and $(V^{+},V^{-})$ form an edge if $U^{+}\cap V^{-}\neq\emptyset$. It is not hard to check that even after the refinements, we still get the right adjoint to $\partial$, i.e., that indeed there is a homomorphism $\mathbf{A}\to\omega_{\partial}(\mathbf{B})$ if and only if there is a homomorphism $\partial(\mathbf{A})\to\mathbf{B}$ for any digraph $\mathbf{A}$ and $\phi$-structure $\mathbf{B}$. Note how this compares to the right adjoint $\delta_{R}$ of the arc digraph functor as defined in the previous subsection.

6.3. Proof of Theorem 6.2

As noted above, the proof closely follows the proof of Theorem 5.2, and we present it with the same structure starting with the easier of the two implications.

Lemma 6.3**.**

If there is a homomorphism $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$ then there is a homomorphism $g\colon\mathbf{A}\to\Omega(\mathbf{B})$.

Proof 6.4**.**

Assuming $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$, we define a mapping $g\colon A\to\Omega(B)$ by setting, for all $u\in A$ and all $t\in\mathscr{T}_{V}$,

[TABLE]

and claim that it is a homomorphism from $\mathbf{A}$ to $\Omega(\mathbf{B})$. We need to check that $g(u)$ is well-defined, i.e., that $f\circ h^{\check{S}}$ is a homomorphism from $\Gamma({\mathbf{T}(t)})$ to $\mathbf{B}$ for all $t\in\mathscr{T}_{V}$ and that $g(u)_{{\operatorname{\sf vertex}}}=\{\emptyset\}$. Indeed, $f\circ h^{\check{S}}$ is a homomorphism since it is a composition of homomorphisms $h^{\check{S}}\colon\Gamma({\mathbf{T}(t)})\to\Gamma(\mathbf{A})$ and $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$. And, $g(u)_{{\operatorname{\sf vertex}}}=\{\emptyset\}$ since ${\mathbf{T}({\operatorname{\sf vertex}})}$ has no ${\check{S}}$-edges which implies that $h^{\check{S}}=\emptyset$ (i.e., $h^{\check{S}}$ is the empty map), and consequently $f\circ h^{\check{S}}=\emptyset$, for the only mapping $h\colon T({\operatorname{\sf vertex}})\to A$ such that $h(r_{\operatorname{\sf vertex}})=u$.

To show that $g$ is a homomorphism, assume first that $S\neq{\check{S}}$ is a $\sigma$-symbol of arity $k$, and $e=(u_{1},\dots,u_{k})\in S^{\mathbf{A}}$. We claim that, for each $t\in\mathscr{T}_{S}$,

[TABLE]

Let $t=\operatorname{\sf edge}_{S}(t_{1},\dots,t_{k})$, and observe that homomorphisms $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ such that $h^{S}(r_{t})=e$ are in 1-to-1 correspondence with $k$-tuples of homomorphisms $h_{1},\dots,h_{k}$, such that $h_{i}\colon{\mathbf{T}(t_{i})}\to\mathbf{A}$ and $h_{i}(r_{t_{i}})=u_{i}$ for all $i\in[k]$, obtained as their restrictions to the respective subtrees. Finally, if $h$ is the union of $h_{i}$’s then also $h^{\check{S}}$ is the union of $h_{i}^{\check{S}}$’s. The claim then easily follows. For (B1), we want to check that $\operatorname{\text{\sf\&}}^{\!S}(g(e))_{t}\subseteq g(u_{i})_{s}$ where $s=\operatorname{\sf pr}_{i}(t)$. Observe that if $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ and $h^{S}(r_{t})=e$ then $h(r_{s})=u_{i}$ since the $i$-th component of $r_{t}$ is $r_{s}$. The condition then follows from the claim. Finally, (B2) follows directly from the claim by the same argument as $g(u)_{t}\subseteq\hom({\mathbf{T}(t)},\mathbf{B})$.

Finally, for the case $S={\check{S}}$ and $e=(u_{1},\dots,u_{k})\in\check{S}^{\mathbf{A}}$, we need to pick an element $w_{\bullet}\in B$ witnessing that $g^{S}(e)\in\check{S}^{\Omega(\mathbf{B})}$ — we pick $w_{\bullet}=f(e)$. Consequently, we have

[TABLE]

for each $\check{S}$-term $t\in\mathscr{T}_{\check{S}}$ such that $t=\operatorname{\sf edge}_{\check{S}}(t_{1},\dots,t_{k})$. We claim that

[TABLE]

The inclusion ‘$\supseteq$’ is clear since the restriction of $h$ to ${\mathbf{T}(t)}_{i}$ is a homomorphism $h_{i}\colon{\mathbf{T}(t_{i})}\to\mathbf{A}$ such that $h_{i}(r_{t_{i}})=u_{i}$. The other inclusion also follows since given homomorphisms $h_{i}\colon{\mathbf{T}(t_{i})}\to\mathbf{A}$ with $h_{i}(r_{t_{i}})=u_{i}$, their union is a homomorphism $h\colon{\mathbf{T}(t)}\to\mathbf{A}$ with $h^{\check{S}}(r_{t})=e$. Moreover,

[TABLE]

and hence

[TABLE]

The conditions (B1) and (B2) then follows in the same way as above.

The other implication is proved in the following two lemmas.

Lemma 6.5**.**

Let $t\in\mathscr{T}$, $h\colon{\mathbf{T}(t)}\to\Omega(\mathbf{B})$, and $d\colon\check{S}^{{\mathbf{T}(t)}}\to B$ be a map such that, for all $s\in\check{S}^{{\mathbf{T}(t)}}$, $d(s)$ is a witness for the edge $h^{\check{S}}(s)$. Then $d$ is a homomorphism $\Gamma({\mathbf{T}(t)})\to\mathbf{B}$.

Proof 6.6**.**

The proof is similar to the proof of Lemma 5.5. We first show by induction on the term $t$ that either $d\in h(r_{t})_{t}$ if $t$ is a $V$-term, or $d\in\operatorname{\text{\sf\&}}^{\!S}(h^{S}(r_{t}))_{t}$ if $t$ is an $S$-term for a symbol $S\neq{\check{S}}$, or $d\in\operatorname{\text{\sf\&}}^{\!\check{S}}(h^{\check{S}}(r_{t}),d(r_{t}))_{t}$ if $t$ is an $\check{S}$-term.

**Case $t={\operatorname{\sf vertex}}$.: **

Since $\check{S}^{{\mathbf{T}({\operatorname{\sf vertex}})}}=\emptyset$, the only map $d\colon\check{S}^{{\mathbf{T}({\operatorname{\sf vertex}})}}\to B$ is the empty map which is in $h(r_{\operatorname{\sf vertex}})_{\operatorname{\sf vertex}}$ by definition.

**Case $t=\operatorname{\sf edge}_{\check{S}}(t_{1},\dots,t_{k})$.: **

Note that restrictions of $h$ to subtrees ${\mathbf{T}(t_{i})}$’s are homomorphisms, hence, for all $i\in\{1,\dots,k\}$, $h(r_{t_{i}})_{t_{i}}$ contains the corresponding restrictions of $d$ by the inductive assumption. The claim then follows from the definition of $\operatorname{\text{\sf\&}}^{\!\check{S}}$ since

[TABLE]

**Case $t=\operatorname{\sf edge}_{S}(t_{1},\dots,t_{k})$ where $S\neq\check{S}$.: **

This is proved in the same way as the above case, with the exception that the last component of the product does not appear since $r_{t}\not\in\check{S}^{{\mathbf{T}(t)}}$.

**Case $t=\operatorname{\sf pr}_{i}(s)$.: **

Since $h$ is a homomorphism from ${\mathbf{T}(t)}={\mathbf{T}(s)}$ to $\mathbf{B}$, we know that either $\operatorname{\text{\sf\&}}^{\!S}(h(r_{s}))_{s}$, if $t$ is an $S$-term for $S\neq\check{S}$, or $\operatorname{\text{\sf\&}}^{\!\check{S}}(h(r_{s}),a(r_{s}))$, if $t$ is an $\check{S}$-term, contains $d$ by the inductive assumption. The claim then follows from (B1).

The lemma then immediately follows either by the definition, or by (B2).

Similarly as in the case of Lemma 5.5, the lemma before provides the adjunction in the special case $\mathbf{A}={\mathbf{T}(t)}$ for $t\in\mathscr{T}$, and we derive the general case from this.

Lemma 6.7**.**

If there is a homomorphism $g\colon\mathbf{A}\to\Omega(\mathbf{B})$, then there is a homomorphism $f\colon\Gamma(\mathbf{A})\to\mathbf{B}$.

Proof 6.8**.**

As before we assume that, for each $\tau$-symbol $R$, $\mathbf{Q}_{R}$ and ${\mathbf{T}(t_{R})}$ are equal and not just isomorphic. We define a mapping $f\colon\Gamma(A)\to B$ by setting, for all $s\in\check{S}^{\mathbf{A}}$, $f(s)=e_{\bullet}$ for some witness $e_{\bullet}$ of the edge $g^{\check{S}}(s)\in\check{S}^{\Omega(\mathbf{B})}$, and claim that this $f$ is a homomorphism from $\Gamma(\mathbf{A})$ to $\mathbf{B}$.

We need to show that $f$ preserves each relation $R$. Assume that $(u_{1},\dots,u_{k})\in R^{\Gamma(\mathbf{A})}$, i.e., there is a homomorphism $h\colon\mathbf{Q}_{R}\to\mathbf{A}$, s.t.,

[TABLE]

The previous lemma applied to the homomorphism $g\circ h\colon{\mathbf{T}(t_{R})}\to\Omega(\mathbf{B})$ in place of $h$ and the map $f\circ h^{\check{S}}$ in place of $d$ implies that $f\circ h^{\check{S}}\colon\Gamma(\mathbf{Q}_{R})\to\mathbf{B}$ is a homomorphism, which in turn implies

[TABLE]

since $(x_{1},\dots,x_{k})\in R^{\Gamma(\mathbf{Q}_{R})}$, as we wanted to show.

This concludes the proof of Theorem 6.2.

7. Composition of adjoints

In this section, we give an example of what can be achieved by composing functors defined in Sections 5 and 6. The power of composing two adjoints to obtain more complicated constructions was observed in [FT15, Section 5] where the authors considered composition of digraph functors with adjoints. This section gives several examples that show that we can obtain adjoints to more digraph functors by composing functors that go outside of the scope of digraphs into general relational structures. Naturally, our constructions also give more adjoints between general relational structures.

We start with a few general observations. The key fact that makes composition of adjoints useful is the following well-known category-theoretical observation.

Lemma 7.1**.**

Assume that $\Lambda_{1},\Gamma_{1}$ and $\Lambda_{2},\Gamma_{2}$ are two pairs of (thin) adjoint functors, then $\Lambda_{1}\circ\Lambda_{2}$ is a left adjoint to $\Gamma_{2}\circ\Gamma_{1}$.

Proof 7.2**.**

The proof is straightforward. We get the following string of equivalences from the two adjoints: $\Lambda_{1}\Lambda_{2}(\mathbf{A})\to\mathbf{B}$ if and only if $\Lambda_{2}(\mathbf{A})\to\Gamma_{1}(\mathbf{B})$ if and only if $\mathbf{A}\to\Gamma_{2}\Gamma_{1}(\mathbf{B})$ for any two structures $\mathbf{A}$ and $\mathbf{B}$ of the right signatures.

We note that a composition of Pultr functors gives a Pultr functor [Pul70]. We include a sketch of a proof of a slightly weaker statement.

Lemma 7.3**.**

Let $\Lambda_{1},\Gamma_{1}$ and $\Lambda_{2},\Gamma_{2}$ be two pairs of left and central Pultr functors such that $\Lambda_{1}\circ\Lambda_{2}$ and $\Gamma_{2}\circ\Gamma_{1}$ are well-defined. Then there is a pair of Pultr functors $\Lambda$ and $\Gamma$ such that $\Lambda(\mathbf{A})$ and $\Lambda_{1}\circ\Lambda_{2}(\mathbf{A})$ are isomorphic for all $\mathbf{A}$, and $\Gamma(\mathbf{B})$ and $\Gamma_{2}\circ\Gamma_{1}(\mathbf{B})$ are homomorphically equivalent for all $\mathbf{B}$.

Proof 7.4** (Proof sketch).**

The template of the composition can be obtained as the $\Lambda_{1}$-image of the Pultr template defining $\Lambda_{2}$ and $\Gamma_{2}$, i.e., if the template of $\Lambda_{2}$ and $\Gamma_{2}$ is composed of structures $\mathbf{P}$ and $\mathbf{Q}_{R}$, and homomorphisms $\epsilon_{i,R}$, then the template of the composition consists of structures $\Lambda_{1}(\mathbf{P})$ and $\Lambda_{1}(\mathbf{Q}_{R})$, and homomorphisms $\epsilon_{i,R}^{\Lambda_{1}}\colon\Lambda_{1}(\mathbf{P})\to\Lambda_{1}(\mathbf{Q}_{R})$ that are induced by $\Lambda_{1}$ from $\epsilon_{i,R}\colon\mathbf{P}\to\mathbf{Q}_{R}$.

The composition $\Lambda_{1}\circ\Lambda_{2}$ is, by defininion, a two step process: In the first step (applying $\Lambda_{2}$), we replace each vertex with a copy of $\mathbf{P}$, and in the second step (applying $\Lambda_{1}$), each of these copies is replaced with a copy of $\Lambda_{1}(\mathbf{P})$. Analogously, an $R$-edge is replaced with a copy of $\Lambda_{1}(\mathbf{Q}_{R})$. It is also not hard to observe that the identification corresponds to the maps $\epsilon_{i,R}^{\Lambda_{1}}$, which concludes that, for all $\mathbf{A}$, $\Lambda_{1}\circ\Lambda_{2}(\mathbf{A})$ is isomorphic to $\Lambda(\mathbf{A})$. We get that $\Gamma_{2}\circ\Gamma_{1}(\mathbf{B})$ and $\Gamma(\mathbf{B})$ are homomorphically equivalent for all $\mathbf{B}$, since both functors are right adjoints to $\Lambda$.

The following theorem is obtained by composing adjoints constructed in Sections 5 and 6. Although it is not an exhaustive list of adjunctions that can be constructed by such compositions, it provides more adjoints to digraph functors on top of those provided in [FT15]. The theorem concerns a relatively general case of Pultr templates where $\mathbf{P}$ is an arbitrary tree. We only require that copies of $\mathbf{P}$ in the respective $\mathbf{Q}_{R}$’s intersect in at most one vertex. We also note that this theorem covers the cases of central Pultr functors whose adjoints are provided by Theorems 5.2 and 6.2.

Theorem 7.5**.**

Assume a $(\sigma,\tau)$-Pultr template with $\mathbf{P}$ and all $\mathbf{Q}_{R}$’s being $\sigma$-trees such that, for each $\tau$-symbol $R$,

1. (1)

$\epsilon_{i,R}$* is injective for all $i\in\{1,\dots,\operatorname{ar}R\}$, and* 2. (2)

for each $i\neq j$, $i,j\in\{1,\dots,\operatorname{ar}R\}$, the images of $\mathbf{P}$ under $\epsilon_{i,R}$ and $\epsilon_{j,R}$ intersect in at most one vertex.

Then the corresponding central Pultr functor $\Gamma$ has a right adjoint.

Proof 7.6**.**

Assume that $P=\{1,\dots,p\}$. The goal is to decompose the functor $\Gamma$ into two Pultr functors $\Gamma_{1}$ and $\Gamma_{2}$. The intermediate step is to construct a structure of a new signature. This new signature $\upsilon$ is obtained from $\sigma$ by adding a new relational symbol $S$ of arity $p$ (the size of the domain of $\mathbf{P}$) while retaining all symbols in $\sigma$.

We define the first functor, $\Gamma_{1}$ that maps $\sigma$-structures to $\upsilon$-structures. Essentially this functor simply adds a new relation $S$ that is defined by $\mathbf{P}$, i.e., $\Gamma_{1}(\mathbf{B})$ is the $\upsilon$-structure with domain $B$, where the relations are defined as

[TABLE]

It is clear that this functor is defined by a $(\sigma,\upsilon)$-Pultr template that satisfies the assumptions of Theorem 5.2.

The second functor, $\Gamma_{2}$ is defined by altering the original Pultr template for $\Gamma$. The new template consists of $\mathbf{P}^{\prime}$ and $\mathbf{Q}^{\prime}_{R}$’s defined as follows. First, $\mathbf{P}^{\prime}=\mathbf{S}_{1}$. For each $\tau$-symbol $R$, we obtain $\mathbf{Q}^{\prime}_{R}$ from $\mathbf{Q}_{R}$ by the following procedure: For each $i\in\{1,\dots,\operatorname{ar}R\}$, remove from $\mathbf{Q}_{R}$ all edges in the image of $\epsilon_{i,R}$, and add the edge $(\epsilon_{i,R}(1),\dots,\epsilon_{i,R}(p))\in S^{\mathbf{Q}^{\prime}_{R}}$. Finally, we let $\epsilon_{i,R}^{\prime}=\epsilon_{i,R}$, for all $i$ and $R$, which is a homomorphism $\mathbf{P}^{\prime}\to\mathbf{Q}^{\prime}_{R}$ since we added the corresponding $S$-edge into $\mathbf{Q}^{\prime}_{R}$. Observe that since $\epsilon_{i,R}$’s are injective, the above procedure does not introduce reflexive tuples, and since the images of $\epsilon_{i,R}$ and $\epsilon_{j,R}$ intersect in at most one vertex for $i\neq j$, this does not introduce cycles into $\mathbf{Q}^{\prime}_{R}$ using $S$-edges, and since all other edges have been removed in the image, it also does not introduce any other cycles. Hence, $\mathbf{Q}^{\prime}_{R}$ is still a tree for each $\tau$-symbol $R$, and therefore Theorem 6.2 applies.

The above two paragraphs show that $\Gamma_{1}$ and $\Gamma_{2}$ have adjoints $\Omega_{1}$ and $\Omega_{2}$. And it is straightforward to check that $\Gamma=\Gamma_{2}\circ\Gamma_{1}$. This concludes that $\Omega_{1}\circ\Omega_{2}$ is a right adjoint to $\Gamma$ by Lemma 7.1.

8. Conclusion

We have studied the problem of characterising central Pultr functors for arbitrary relational structures that admit a right adjoint, and, for those that do, giving an explicit construction for such an adjoint. There is a necessary condition for the existence of such an adjoint (cf. Theorem 2.9 and comments after it). We gave a sufficient condition in Theorem 7.5. These two conditions do not match, there is a gap between them, and it is not quite clear what the necessary and sufficient condition should be (even in the case of digraphs). Apart from the requirement that $\mathbf{P}$ and all $\mathbf{Q}_{R}$’s are trees, Theorem 7.5 has two additional assumptions. We believe that the second assumption (about intersection of images of $\mathbf{P}$ in $\mathbf{Q}_{R}$) is a technicality that can be removed with some extra work. How essential is the first assumption (about injectivity of homomorphisms $\epsilon_{i,R}$)? For example, is it true that for every central Pultr functor $\Gamma$ that has a right adjoint, there is another central Pultr functor $\Gamma^{\prime}$ such that (a) for every structure $\mathbf{A}$ of appropriate signature, $\Gamma(\mathbf{A})$ and $\Gamma^{\prime}(\mathbf{A})$ are homomorphically equivalent, and (b) the Pultr template corresponding to $\Gamma^{\prime}$ has all homomorphisms $\epsilon_{i,R}$ injective?

Finally, let us discuss possible applications of our results in the complexity of (promise) CSPs. Left and central Pultr functors can alternatively be described as “gadget replacements” and “pp-constructions”, respectively. These two constructions are central to the algebraic theory of complexity of (promise) CSPs where gadget replacements are used as log-space reductions between (promise) CSPs whose templates are related via pp-constructions (see, e.g., [KOWŽ23, Section 4.1] for a detailed exposition). The opposite type of reductions, where gadgets are replaced by individual constraints, is less common, but has already been shown to be important; see, e.g., [WŽ20, BK22]. It was discovered recently, that the right adjoints to some functor can be used to characterise when the functor is a valid reduction between two (promise) CSPs [KOWŽ23, Theorem 4.6]. Moreover, recent developments in the theory of promise CSPs have called for characterisation of reductions which include central Pultr functors [BK22, KO22, DO23]. Our results provide a characterisation of when a reduction from a general class is a valid reduction between two promise CSPs. Although, we are not aware of a specific interesting case of a promise CSP whose hardness is proven by providing a right adjoint to a central Pultr functor, apart from the use of the arc graph construction in [WŽ20], our general results can be used as a foundation for further investigation of these reductions.

Acknowledgements

We would like to thank the reviewers for many insightful comments on the manuscripts.